{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Arial Narrow" 1 12 128 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2d input" -1 256 "Arial Narrow" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Helvetica" 0 1 0 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 308 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "Cou rier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 319 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "Cour ier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 330 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 331 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 332 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 335 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 336 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 340 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 341 "Times" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 342 " Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "Times" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 347 "Helvetica" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 348 "Helvetica" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 350 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 352 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE " " -1 353 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 354 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 355 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 356 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 357 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 358 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 361 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 362 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 364 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 365 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 366 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 367 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 368 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 369 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 370 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 371 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 372 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 373 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 374 "Times " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 375 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 376 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 377 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 378 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 379 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 380 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 381 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 382 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 383 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 384 " Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 385 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 386 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 387 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 388 "Helvetica" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 389 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 390 "" 0 1 0 0 64 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 391 "Helvetica" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 392 " " 0 1 111 0 7 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 393 "" 0 1 111 0 7 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 394 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 395 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 396 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 397 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 398 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 399 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 400 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 401 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 402 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 403 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 404 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 405 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 406 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 407 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 408 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 409 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 410 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 411 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 412 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 413 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 414 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 415 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 416 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 417 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 418 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 419 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 420 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 421 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 422 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 423 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 424 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 425 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 426 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 427 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 428 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 429 "Times " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 430 "Times" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 431 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 432 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 433 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 434 "Times" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 435 "Helv etica" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 436 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 437 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 438 "Helvetica" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 439 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 440 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 441 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 442 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 443 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 444 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 445 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 446 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 447 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 448 "Times " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 449 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 450 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 451 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 452 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 453 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 454 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 455 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 456 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 457 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 458 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 459 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 460 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 461 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 462 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 463 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 464 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 465 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 466 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Arial Narrow " 0 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Arial Narrow" 0 12 128 0 128 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 128 128 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 228 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 74 0 52 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE " " -1 -1 "" 0 1 4 0 0 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 274 "" 0 "" {TEXT 388 39 "MATRICES AND MATRIX OPE RATIONS: Unit 19" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 275 "" 0 "" {TEXT 390 23 "Dr. Wlodzislaw Kostecki" }}{PARA 276 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 277 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 278 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 279 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 280 "" 0 "" {TEXT 389 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 281 "" 0 "" {TEXT 391 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 282 "" 0 "" {TEXT 392 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(19)" }{TEXT 348 1 " " }{TEXT 347 76 "Solution of syste ms of linear algebraic inhomogeneous equations using Cramer" }{TEXT 434 1 "\222" }{TEXT 435 6 "s rule" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 438 10 "OBJECTIVES" }{TEXT 439 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 440 1 "\225" }{TEXT -1 65 " To define a system of linear algebr aic inhomogeneous equations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 441 1 " \225" }{TEXT -1 75 " To show how a system of such equations may be wr itten in the matrix form." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 443 1 "\225 " }{TEXT -1 36 " To introduce the concepts of the " }{TEXT 442 18 "c oefficient matrix" }{TEXT -1 3 ", " }{TEXT 444 15 "solution vector" } {TEXT -1 8 ", and " }{TEXT 445 20 "inhomogeneous vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 448 1 "\225" }{TEXT -1 16 " To \+ introduce " }{TEXT 446 7 "Cramer\222" }{TEXT -1 1 "s" }{TEXT 447 2 " \+ " }{TEXT -1 69 "rule for solving systems of linear algebraic inhomoge neous equations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 450 1 "\225" }{TEXT -1 57 " To provide a numerical example for the application of " } {TEXT 449 7 "Cramer\222" }{TEXT -1 8 "s rule." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 452 1 "\225" }{TEXT -1 102 " To show how systems of linear algebraic inhomogeneous equations may be solved in a simple way using " }{TEXT 454 5 "Maple" }{TEXT 453 1 "\222" }{TEXT -1 12 "s functions \+ " }{TEXT 455 8 "linsolve" }{TEXT -1 4 " or " }{TEXT 456 5 "solve" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 451 1 "\225" }{TEXT -1 33 " To demonstrate that so-called " }{TEXT 457 15 "ill-conditioned " }{TEXT -1 81 " systems of linear inhomogeneous equations do not pos e problems if tackled with " }{TEXT 458 5 "Maple" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "restart : with(linalg, copyinto, det, genmatrix, li nsolve, multiply, stackmatrix) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Consider the following sys tem of " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 47 " linear algebrai c inhomogeneous equations in " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 12 " variables:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "a[11]*x[1]+a[12]*x[2]+a[13]*x[ 3]+a[14]*x[4]=k[1]" "6#/,**&&%\"aG6#\"#6\"\"\"&%\"xG6#F*F*F**&&F'6#\"# 7F*&F,6#\"\"#F*F**&&F'6#\"#8F*&F,6#\"\"$F*F**&&F'6#\"#9F*&F,6#\"\"%F*F *&%\"kG6#F*" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {XPPEDIT 18 0 "a[21]*x[1]+a[22]*x[2]+a[23]*x[3]+a[24]*x[4]=k[2]" "6 #/,**&&%\"aG6#\"#@\"\"\"&%\"xG6#F*F*F**&&F'6#\"#AF*&F,6#\"\"#F*F**&&F' 6#\"#BF*&F,6#\"\"$F*F**&&F'6#\"#CF*&F,6#\"\"%F*F*&%\"kG6#F4" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {XPPEDIT 18 0 "a[3 1]*x[1]+a[32]*x[2]+a[33]*x[3]+a[34]*x[4]=k[3]" "6#/,**&&%\"aG6#\"#J\" \"\"&%\"xG6#F*F*F**&&F'6#\"#KF*&F,6#\"\"#F*F**&&F'6#\"#LF*&F,6#\"\"$F* F**&&F'6#\"#MF*&F,6#\"\"%F*F*&%\"kG6#F;" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {XPPEDIT 18 0 "a[41]*x[1]+a[42]*x[2]+a[ 43]*x[3]+a[44]*x[4]=k[4]" "6#/,**&&%\"aG6#\"#T\"\"\"&%\"xG6#F*F*F**&&F '6#\"#UF*&F,6#\"\"#F*F**&&F'6#\"#VF*&F,6#\"\"$F*F**&&F'6#\"#WF*&F,6#\" \"%F*F*&%\"kG6#FB" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "where the term " }{TEXT 258 13 "inhomoge neous" }{TEXT -1 24 " denotes that at least " }{TEXT 259 3 "one" } {TEXT -1 17 " of the numbers " }{XPPEDIT 18 0 "k[1]" "6#&%\"kG6#\"\" \"" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 "k[4]" "6#&%\"kG6#\"\"%" } {TEXT -1 5 " is " }{TEXT 260 3 "not" }{TEXT -1 2 " " }{XPPMATH 20 "6 #%%zeroG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Since the number of variables equa ls the number of equations of this system, it is said to be a " } {TEXT 413 6 "square" }{TEXT -1 29 " system of linear equations." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "This system of equations may be written in the matrix form. To \+ this end, do the following." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 307 1 "\225" }{TEXT -1 14 " Define the " }{TEXT 261 6 "square" }{TEXT -1 34 " matrix containing coefficien ts " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 10 " as the \+ " }{TEXT 262 18 "coefficient matrix" }{TEXT -1 3 " [" }{TEXT 263 1 " A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "A := matrix(4, 4, [ a[11], a[12], a[13], a[14], a[21], a[22], a[23], a[24] , a[31], a[32], a[33], a[34], a[41], a[42], a[43], a[44] ]) : A = ma trix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&&% \"aG6#\"#6&F+6#\"#7&F+6#\"#8&F+6#\"#97&&F+6#\"#@&F+6#\"#A&F+6#\"#B&F+6 #\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M7&&F+6#\"#T&F+6#\"#U&F+6#\" #V&F+6#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 1 "\225" }{TEXT -1 14 " Define the " }{TEXT 264 13 "column matrix" }{TEXT -1 24 " containing variables " }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 10 " as the " }{TEXT 265 15 "solution vector" }{TEXT -1 3 " [" }{TEXT 266 1 "X" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "X := matrix(4, 1, [ x[1], x[2], x[3], x[4] ]) : X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"XG-%'matrixG6#7&7#&%\"xG6#\"\"\"7#&F+6#\"\"#7#&F+6#\"\"$7#&F +6#\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 309 1 "\225" }{TEXT -1 14 " Define the " }{TEXT 267 13 "column matrix" }{TEXT -1 22 " containing numbers " }{XPPEDIT 18 0 " k[i]" "6#&%\"kG6#%\"iG" }{TEXT -1 10 " as the " }{TEXT 268 20 "inhom ogeneous vector" }{TEXT -1 3 " [" }{TEXT 269 1 "K" }{TEXT -1 2 "]:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "K := matrix(4, 1, [ k[1], \+ k[2], k[3], k[4] ]) : K = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'matrixG6#7&7#&%\"kG6#\"\"\"7#&F+6#\"\"#7#&F+6#\"\"$7#&F +6#\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 330 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 328 4 "N.B." }{TEXT -1 61 " Both the \" solution vector\" and \"inhomogeneous vector\" are " }{TEXT 329 6 "co lumn" }{TEXT -1 55 " structures and as such, cannot be named \"vector s\" in " }{TEXT 332 5 "Maple" }{TEXT -1 121 " \226 refer to Unit (5). \+ However, because of a customary practice used in textbooks, both names are maintained in this Unit." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 270 "" 0 "" {TEXT 331 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "With the \+ matrices defined as above, the system of the equations may now be writ ten in the matrix form" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 1 "[" } {TEXT 270 1 "A" }{TEXT -1 3 "] [" }{TEXT 271 1 "X" }{TEXT -1 1 "]" } {TEXT 382 3 " = " }{TEXT -1 1 "[" }{TEXT 272 1 "K" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "or," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "matrix(A) * ma trix(X) = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG 6#7&7&&%\"aG6#\"#6&F+6#\"#7&F+6#\"#8&F+6#\"#97&&F+6#\"#@&F+6#\"#A&F+6# \"#B&F+6#\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M7&&F+6#\"#T&F+6#\"# U&F+6#\"#V&F+6#\"#W\"\"\"-F&6#7&7#&%\"xG6#Fhn7#&F^o6#\"\"#7#&F^o6#\"\" $7#&F^o6#\"\"%Fhn-F&6#7&7#&%\"kGF_o7#&FapFbo7#&FapFfo7#&FapFjo" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Provided that the matrix [" }{TEXT 274 1 "A" }{TEXT -1 6 "] is \+ " }{TEXT 273 12 "non-singular" }{TEXT -1 27 ", i.e. its determinant \+ is " }{TEXT 276 3 "not" }{TEXT -1 2 " " }{XPPMATH 20 "6#%%zeroG" } {TEXT -1 24 ", it follows that the " }{TEXT 275 7 "inverse" }{TEXT -1 17 " to the matrix [" }{TEXT 277 1 "A" }{TEXT -1 57 "] exists. Mul tiplying each side of the matrix equation [" }{TEXT 363 1 "A" }{TEXT -1 3 "] [" }{TEXT 364 1 "X" }{TEXT -1 1 "]" }{TEXT 383 3 " = " }{TEXT -1 1 "[" }{TEXT 365 1 "K" }{TEXT -1 7 "] by " }{TEXT 366 3 "Inv" } {TEXT -1 1 "[" }{TEXT 367 1 "A" }{TEXT -1 9 "] yields" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 368 4 "(Inv" }{TEXT -1 1 "[" }{TEXT 369 1 "A" }{TEXT -1 1 "]" }{TEXT 371 1 ")" }{TEXT -1 2 " [" }{TEXT 370 1 "A" } {TEXT -1 3 "] [" }{TEXT 278 1 "X" }{TEXT -1 1 "]" }{TEXT 384 7 " = (In v" }{TEXT -1 1 "[" }{TEXT 279 1 "A" }{TEXT -1 1 "]" }{TEXT 372 1 ")" } {TEXT -1 2 " [" }{TEXT 280 1 "K" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Since " } {TEXT 374 4 "(Inv" }{TEXT -1 1 "[" }{TEXT 375 1 "A" }{TEXT -1 1 "]" } {TEXT 377 1 ")" }{TEXT -1 2 " [" }{TEXT 376 1 "A" }{TEXT -1 1 "]" } {TEXT 385 3 " = " }{TEXT -1 1 "[" }{TEXT 373 1 "U" }{TEXT -1 25 "] [s ee Unit (14)] and [" }{TEXT 379 1 "U" }{TEXT -1 3 "] [" }{TEXT 380 1 "X" }{TEXT -1 1 "]" }{TEXT 386 3 " = " }{TEXT -1 1 "[" }{TEXT 381 1 "X " }{TEXT -1 54 "] [see Unit (9)], the above simplifies to yield the \+ " }{TEXT 362 15 "solution vector" }{TEXT -1 13 " in the form" }}} {EXCHG {PARA 273 "" 0 "" {TEXT -1 1 "[" }{TEXT 359 1 "X" }{TEXT -1 1 " ]" }{TEXT 387 7 " = (Inv" }{TEXT -1 1 "[" }{TEXT 360 1 "A" }{TEXT -1 1 "]" }{TEXT 378 1 ")" }{TEXT -1 2 " [" }{TEXT 361 1 "K" }{TEXT -1 1 " ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "matrix(X) = Inv(matrix(A)) * matrix(K) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7&7#&%\"xG6#\"\"\"7#&F*6# \"\"#7#&F*6#\"\"$7#&F*6#\"\"%*&-%$InvG6#-F%6#7&7&&%\"aG6#\"#6&FB6#\"#7 &FB6#\"#8&FB6#\"#97&&FB6#\"#@&FB6#\"#A&FB6#\"#B&FB6#\"#C7&&FB6#\"#J&FB 6#\"#K&FB6#\"#L&FB6#\"#M7&&FB6#\"#T&FB6#\"#U&FB6#\"#V&FB6#\"#WF,-F%6#7 &7#&%\"kGF+7#&FdpF/7#&FdpF37#&FdpF7F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 325 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 4 "N.B." }{TEXT -1 10 " When a " }{TEXT 316 15 "solution vector " }{TEXT -1 3 " [" }{TEXT 315 1 "X" }{TEXT -1 48 "] exists whose elem ents simultaneously satisfy " }{TEXT 324 3 "all" }{TEXT -1 61 " the \+ equations in the system, the equations are said to be " }{TEXT 317 10 "consistent" }{TEXT -1 89 ". If no solution vector exists having t his property, then the equations are said to be " }{TEXT 318 12 "inco nsistent" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 268 "" 0 "" {TEXT 326 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "A useful method of solution for " }{TEXT 312 5 "small" }{TEXT -1 37 " systems of such equations (maximum " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 74 " eq uations where computations are to be performed manually) is known as \+ " }{TEXT 281 7 "Cramer\222" }{TEXT -1 1 "s" }{TEXT 349 2 " " }{TEXT -1 34 "rule. It states that any unknown " }{XPPEDIT 18 0 "x[i]" "6#&% \"xG6#%\"iG" }{TEXT -1 13 " is found as" }}}{EXCHG {PARA 272 "" 0 "" {XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT 334 8 " = Det([" }{TEXT 335 1 "A" }{TEXT 336 3 ", " }{TEXT 341 1 "j" }{TEXT 342 4 " <\226 " } {TEXT 337 1 "K" }{TEXT 338 2 "])" }{TEXT 343 1 "/" }{TEXT 344 5 "Det([ " }{TEXT 339 1 "A" }{TEXT 340 2 "])" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The determinant in the \+ " }{TEXT 345 9 "numerator" }{TEXT -1 40 " is defined as the determin ant of the " }{TEXT 283 8 "modified" }{TEXT -1 22 " coefficient matr ix [" }{TEXT 282 1 "A" }{TEXT -1 17 "], in which the " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 51 "th column is replaced by the inhomogeneo us vector [" }{TEXT 284 1 "K" }{TEXT -1 33 "], as indicated by the not ation " }{TEXT 425 1 "[" }{TEXT 426 1 "A" }{TEXT 427 3 ", " }{TEXT 430 1 "j" }{TEXT 431 4 " <\226 " }{TEXT 428 1 "K" }{TEXT 429 1 "]" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Exemplarily, the numerator for the unknow n " }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }{TEXT -1 14 " has the f orm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "i := 1 : j := 2 : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "`det(A, j <\226 K)` := \+ copyinto(K, A, i, j) : Det(A, ` j <\226 K`) = Det(matrix(`det(A, j < \226 K)`)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%(~j~<| at~KG-F%6#-%'matrixG6#7&7&&%\"aG6#\"#6&%\"kG6#\"\"\"&F16#\"#8&F16#\"#9 7&&F16#\"#@&F56#\"\"#&F16#\"#B&F16#\"#C7&&F16#\"#J&F56#\"\"$&F16#\"#L& F16#\"#M7&&F16#\"#T&F56#\"\"%&F16#\"#V&F16#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "In the denom inator, the determinant of the coefficient matrix [" }{TEXT 286 1 "A" }{TEXT -1 17 "] is called the " }{TEXT 285 26 "characteristic determi nant" }{TEXT -1 69 ". For this particular case, it is given by the fo llowing expression:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det (A)` := det(A) : Det(A) = `det(A)` ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG,R**&%\"aG6#\"#6\"\"\"&%\"kG6#\"\"#F.&F+6#\"#LF.& F+6#\"#WF.F.**F*F.F/F.&F+6#\"#MF.&F+6#\"#VF.!\"\"**F*F.&F06#\"\"$F.&F+ 6#\"#BF.F6F.F@**F*F.FBF.&F+6#\"#CF.F=F.F.**F*F.&F06#\"\"%F.FEF.F:F.F.* *F*F.FMF.FIF.F3F.F@**&F+6#\"#@F.&F06#F.F.F3F.F6F.F@**FRF.FUF.F:F.F=F.F .**FRF.FBF.&F+6#\"#8F.F6F.F.**FRF.FBF.&F+6#\"#9F.F=F.F@**FRF.FMF.FYF.F :F.F@**FRF.FMF.FgnF.F3F.F.**&F+6#\"#JF.FUF.FEF.F6F.F.**F]oF.FUF.FIF.F= F.F@**F]oF.F/F.FYF.F6F.F@**F]oF.F/F.FgnF.F=F.F.**F]oF.FMF.FYF.FIF.F.** F]oF.FMF.FgnF.FEF.F@**&F+6#\"#TF.FUF.FEF.F:F.F@**FfoF.FUF.FIF.F3F.F.** FfoF.F/F.FYF.F:F.F.**FfoF.F/F.FgnF.F3F.F@**FfoF.FBF.FYF.FIF.F@**FfoF.F BF.FgnF.FEF.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 17 "Numerical example" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Solve the fo llowing system of " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 19 " equa tions using " }{TEXT 350 7 "Cramer\222" }{TEXT -1 8 "s rule:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Eq[1] := x[1] - 3*x[2] \+ + x[3] - x[4] = -3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Eq[2] := 2*x[1] - x[2] + 3*x[3] + x[4] = 2 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Eq[3] := x[1] + x[2] + 2*x[3] + 2*x[4] = 3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Eq[4] := x[1] - 2*x[2] + 4*x[3] + x [4] = 0 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Eq[1] ; Eq[ 2] ; Eq[3] ; Eq[4] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6 #\"\"\"F(*&\"\"$F(&F&6#\"\"#F(!\"\"&F&6#F*F(&F&6#\"\"%F.!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6#\"\"\"\"\"#&F&6#F)!\"\"*&\"\"$F(& F&6#F.F(F(&F&6#\"\"%F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6 #\"\"\"F(&F&6#\"\"#F(*&F+F(&F&6#\"\"$F(F(*&F+F(&F&6#\"\"%F(F(F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"xG6#\"\"\"F(*&\"\"#F(&F&6#F*F(! \"\"*&\"\"%F(&F&6#\"\"$F(F(&F&6#F/F(\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 432 6 "Step 1" }{TEXT -1 65 ". Assign name and value to the number of equations and variable s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "No_Eq := 4 : No_x : = 4 : 'No_Eq' = No_Eq ; 'No_x' = No_x ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&No_EqG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%No _xG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 288 6 "Step 2" }{TEXT -1 17 ". Construct the " }{TEXT 289 18 "coefficient matrix" }{TEXT -1 3 " [" }{TEXT 290 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 459 8 "Method 1" }{TEXT -1 21 ". Using the function " } {TEXT 460 9 "genmatrix" }{TEXT -1 85 " and either of the two alternati ve methods of specifying the equations and variables:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 461 1 " \225" }{TEXT -1 131 " enter manually the names of all equations enclo sed in brackets, followed by the names of all variables enclosed in br ackets, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "A := genmat rix([Eq[1], Eq[2], Eq[3], Eq[4]], [x[1], x[2], x[3], x[4]]) : A = ma trix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&\" \"\"!\"$F*!\"\"7&\"\"#F,\"\"$F*7&F*F*F.F.7&F*!\"#\"\"%F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 462 1 " \225" }{TEXT -1 107 " create a bracketed sequence (list) of equations , followed by bracketed sequence (list) of variables, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "`seq(Eq[i])` := [seq(Eq[i], i=1..No _Eq)] : `seq(x[i])` := [seq(x[i], i=1..No_x)] :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 61 "A := genmatrix(`seq(Eq[i])`, `seq(x[i])`) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6 #7&7&\"\"\"!\"$F*!\"\"7&\"\"#F,\"\"$F*7&F*F*F.F.7&F*!\"#\"\"%F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 8 "Method 2" }{TEXT -1 34 ". Requiring several steps and the " } {TEXT 356 3 "for" }{TEXT -1 16 "-loop construct:" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "(a) Count \+ the number of operands of the left-hand side of each equation:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "for j to No_Eq do if op(se lect(type, convert(lhs(Eq[j]), list), numeric)) <> NULL then ops(lhs _Eq[j]) := 1 else ops(lhs_Eq[j]) := nops(lhs(Eq[j])) fi : print(N o_operands(lhs_Eq[j]) = ops(lhs_Eq[j])) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,No_operandsG6#&%'lhs_EqG6#\"\"\"\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%,No_operandsG6#&%'lhs_EqG6#\"\"#\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,No_operandsG6#&%'lhs_EqG6#\"\"$\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,No_operandsG6#&%'lhs_EqG6# \"\"%F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "(b) Create lists of operands of the left-hand side o f each equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "for j \+ to No_Eq do if ops(lhs_Eq[j]) = 1 then L[j] := [lhs(Eq[j])] else L[ j] := convert(lhs(Eq[j]), list) fi : print(evaln(L[j]) = L[j]) : \+ od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"\"7&&%\"xGF&,$&F* 6#\"\"#!\"$&F*6#\"\"$,$&F*6#\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"#7&,$&%\"xG6#\"\"\"F',$&F+F&!\"\",$&F+6#\"\"$F4&F+6 #\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"$7&&%\"xG6#\"\" \"&F*6#\"\"#,$&F*F&F/,$&F*6#\"\"%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"LG6#\"\"%7&&%\"xG6#\"\"\",$&F*6#\"\"#!\"#,$&F*6#\"\"$F'&F*F&" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "(c) Construct row matrices containing numeric operands of the e lements of each list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "f or i to No_x do n := 0 : for j to No_Eq do if has(L[i], x[j]) = fa lse then a[i,j] := 0 else n := n+1 : a[i,j] := op(n, L[i])/x[j] \+ fi : od : RM[i] := matrix(1, No_x, [[seq(a[i,m], m=1..No_x)]]) : \+ print(RM[i] = matrix(RM[i])) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#RMG6#\"\"\"-%'matrixG6#7#7&F'!\"$F'!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#RMG6#\"\"#-%'matrixG6#7#7&F'!\"\"\"\"$\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#RMG6#\"\"$-%'matrixG6#7#7&\"\"\"F- \"\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#RMG6#\"\"%-%'matrixG6#7 #7&\"\"\"!\"#F'F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "(d) Create a sequence of the row matrices :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`seq(RM[i])` := seq(ma trix(RM[k]), k=1..No_Eq) : sequence(RM['i']) = `seq(RM[i])` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)sequenceG6#&%#RMG6#%\"iG6&-%'matri xG6#7#7&\"\"\"!\"$F1!\"\"-F-6#7#7&\"\"#F3\"\"$F1-F-6#7#7&F1F1F8F8-F-6# 7#7&F1!\"#\"\"%F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "(e) Construct matrix [" }{TEXT 463 1 "A" }{TEXT -1 31 "] by stacking the row matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A := stackmatrix(`seq(RM[i])`) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&\"\"\"!\"$ F*!\"\"7&\"\"#F,\"\"$F*7&F*F*F.F.7&F*!\"#\"\"%F*" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 464 4 "N.B." } {TEXT -1 61 " Method 2 is universal and works even if some coefficien ts " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 37 " in the \+ simultaneous equations are " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 25 ". It also works even if " }{TEXT 465 3 "one" }{TEXT -1 20 " coefficien t is non-" }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 80 " in any equation (t he case purely abstractive from mathematical point of view)." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If " }{TEXT 466 4 "none" }{TEXT -1 23 " of the coefficients \+ " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 6 " is " } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 58 ", the following short variant o f this method may be used:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a := matrix(No_x, No_Eq) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for i to No_Eq do for \+ j to No_x do a[i,j] := op(j, lhs(Eq[i]))/x[j] od : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := matrix(a) : A = matri x(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7&\"\"\" !\"$F*!\"\"7&\"\"#F,\"\"$F*7&F*F*F.F.7&F*!\"#\"\"%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 433 6 "Step 3 " }{TEXT -1 27 ". Create a copy of matrix [" }{TEXT 419 1 "A" }{TEXT -1 33 "] for the future use (in Step 7):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "C_A := copy(A) : C_A = matrix(C_A) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%$C_AG-%'matrixG6#7&7&\"\"\"!\"$F*!\"\"7&\"\"#F ,\"\"$F*7&F*F*F.F.7&F*!\"#\"\"%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 6 "Step 4" }{TEXT -1 24 ". De fine and input the " }{TEXT 291 15 "solution vector" }{TEXT -1 3 " [ " }{TEXT 292 1 "X" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 351 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 357 3 "for" }{TEXT -1 16 "-loop construct:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "x := array(1..No_Eq) : fo r i to No_x do x[i] := x[i] od : X := convert(x, matrix) : X = m atrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"XG-%'matrixG6#7&7#&% \"xG6#\"\"\"7#&F+6#\"\"#7#&F+6#\"\"$7#&F+6#\"\"%" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 352 8 "Method 2" }{TEXT -1 48 ". Using manual inputting of each vector element:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "X := matrix(4, 1, [x[1], x[2 ], x[3], x[4]]) : X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%\"XG-%'matrixG6#7&7#&%\"xG6#\"\"\"7#&F+6#\"\"#7#&F+6#\"\"$7#&F+6#\" \"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 294 6 "Step 5" }{TEXT -1 24 ". Define and input the " }{TEXT 295 20 "inhomogeneous vector" }{TEXT -1 3 " [" }{TEXT 296 1 "K" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 353 8 "Method 1" }{TEXT -1 12 ". Using the " } {TEXT 358 3 "for" }{TEXT -1 16 "-loop construct:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "k := array(1..No_Eq) : for i to No_x do k [i] := rhs(Eq[i]) od : K := convert(k, matrix) : K = matrix(K) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'matrixG6#7&7#!\"$7#\"\"#7 #\"\"$7#\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 354 8 "Method 2" }{TEXT -1 48 ". Using manual in putting of each vector element:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "K := matrix(4, 1, [rhs(Eq[1]), rhs(Eq[2]), rhs(Eq[3]), rhs(Eq[ 4])]) : K = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%' matrixG6#7&7#!\"$7#\"\"#7#\"\"$7#\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The system of the equations may now be written in the matrix form, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "matrix(A) * matrix(X) = matrix(K) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7&7&\"\"\"!\"$F*!\" \"7&\"\"#F,\"\"$F*7&F*F*F.F.7&F*!\"#\"\"%F*F*-F&6#7&7#&%\"xG6#F*7#&F96 #F.7#&F96#F/7#&F96#F3F*-F&6#7&7#F+7#F.7#F/7#\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 6 "Step 6" } {TEXT -1 15 ". Compute the " }{TEXT 297 26 "characteristic determinan t" }{TEXT -1 29 " of the coefficient matrix [" }{TEXT 287 1 "A" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` \+ := det(A) : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$DetG6#%\"AG!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 6 "Step 7" }{TEXT -1 21 ". Find the unknowns " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 5 " ... " } {XPPEDIT 18 0 "x[4]" "6#&%\"xG6#\"\"%" }{TEXT -1 2 " :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "( a) Finding " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`det(A, 1 <\226 K)` := copyinto(K, \+ A, 1, 1) : Det(A, `1 <\226 K`) = Det(matrix(`det(A, 1 <\226 K)`)) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'1~<|at~KG-F%6#-%' matrixG6#7&7&!\"$F0\"\"\"!\"\"7&\"\"#F2\"\"$F17&F5F1F4F47&\"\"!!\"#\" \"%F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`det(A, 1 <\226 K)` := det(`det(A, 1 <\226 K)`) \+ : Det(A, `1 <\226 K`) = `det(A, 1 <\226 K)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'1~<|at~KG!\"(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x[1] := \+ `det(A, 1 <\226 K)`/`det(A)` : 'x[1]' = x[1] ; x[1] := 'x[1]' :" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(b) Findi ng " }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix(C_A) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`det(A, 2 <\226 K)` := copyinto(K, A, 1, 2) : Det(A , `2 <\226 K`) = Det(matrix(`det(A, 2 <\226 K)`)) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$DetG6$%\"AG%'2~<|at~KG-F%6#-%'matrixG6#7&7&\"\"\" !\"$F0!\"\"7&\"\"#F4\"\"$F07&F0F5F4F47&F0\"\"!\"\"%F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`det(A, 2 <\226 K)` := det(`det(A, 2 <\226 K)`) : Det(A, `2 < \226 K`) = `det(A, 2 <\226 K)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$DetG6$%\"AG%'2~<|at~KG!#9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x[2] := `det(A, 2 <\226 K) `/`det(A)` : 'x[2]' = x[2] ; x[2] := 'x[2]' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(c) Finding " }{XPPEDIT 18 0 "x[3]" "6#&%\"xG6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix(C_A) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "` det(A, 3 <\226 K)` := copyinto(K, A, 1, 3) : Det(A, `3 <\226 K`) = D et(matrix(`det(A, 3 <\226 K)`)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$DetG6$%\"AG%'3~<|at~KG-F%6#-%'matrixG6#7&7&\"\"\"!\"$F1!\"\"7&\"\"# F2F4F07&F0F0\"\"$F47&F0!\"#\"\"!F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`det(A, 3 <\226 K )` := det(`det(A, 3 <\226 K)`) : Det(A, `3 <\226 K`) = `det(A, 3 < \226 K)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'3~<|at~ KG!\"(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "x[3] := `det(A, 3 <\226 K)`/`det(A)` : 'x[3] ' = x[3] ; x[3] := 'x[3]' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" xG6#\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(d) Finding " }{XPPEDIT 18 0 "x[4]" "6#& %\"xG6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix (C_A) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`det(A, 4 <\226 \+ K)` := copyinto(K, A, 1, 4) : Det(A, `4 <\226 K`) = Det(matrix(`det( A, 4 <\226 K)`)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG% '4~<|at~KG-F%6#-%'matrixG6#7&7&\"\"\"!\"$F0F17&\"\"#!\"\"\"\"$F37&F0F0 F3F57&F0!\"#\"\"%\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`det(A, 4 <\226 K)` := det(` det(A, 4 <\226 K)`) : Det(A, `4 <\226 K`) = `det(A, 4 <\226 K)` ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'4~<|at~KG\"\"(" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x[4] := `det(A, 4 <\226 K)`/`det(A)` : 'x[4]' = x[4 ] ; x[4] := 'x[4]' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\" \"%!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 310 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 305 4 "N.B." }{TEXT -1 23 " The solutio n vector [" }{TEXT 306 1 "X" }{TEXT -1 23 "] may be obtained in a " } {TEXT 319 7 "simpler" }{TEXT -1 15 " way using the " }{TEXT 304 8 "lin solve" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "A := genmatrix(`seq(Eq[i])`, `seq(x[i])`) : X := li nsolve(A, K) : X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"XG-%'matrixG6#7&7#\"\"\"7#\"\"#F)7#!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Extracting the nu merical value of each unknown from the solution vector [" }{TEXT 417 1 "X" }{TEXT -1 7 "] gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "for i to No_Eq do x[i] := X[i,1] : print(evaln(x[i]) = x[i]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"xG6#\"\"%!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 265 "" 0 "" {TEXT 320 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Thus, the " } {TEXT 321 6 "solved" }{TEXT -1 71 " system of the equations may be wr itten in the matrix form as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "matrix(A) * matrix(X) = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7&7&\"\"\"!\"$F*!\"\"7&\"\"#F,\"\"$F* 7&F*F*F.F.7&F*!\"#\"\"%F*F*-F&6#7&7#F*7#F.F77#F,F*-F&6#7&7#F+F87#F/7# \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Verification of the solution may be performed by evalu ating the product of matrices [" }{TEXT 302 1 "A" }{TEXT -1 7 "] and [ " }{TEXT 303 1 "X" }{TEXT -1 7 "], viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AX` := multiply(A, X) : A * X = matrix(`AX`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"XGF&-%'matrixG6#7&7#! \"$7#\"\"#7#\"\"$7#\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "which is equal to the " }{TEXT 322 20 "inhomogeneous vector" }{TEXT -1 3 " [" }{TEXT 323 1 "K" } {TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 311 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 300 4 "N.B." }{TEXT -1 6 " Th e " }{TEXT 346 8 "simplest" }{TEXT -1 48 " method of solving systems o f such equations in " }{TEXT 333 5 "Maple" }{TEXT -1 17 " is by using \+ the " }{TEXT 313 5 "solve" }{TEXT -1 5 " (or " }{TEXT 301 6 "fsolve" } {TEXT -1 95 ") function. For instructive purposes, this is done hereun der for the above system of equations." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for i to No _x do x[i] := evaln(x[i]) : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "solution := solve(\{Eq[1], Eq[2], Eq[3], Eq[4]\}, \{x [1], x[2], x[3], x[4]\}) : solution ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/&%\"xG6#\"\"$\"\"\"/&F&6#\"\"#F-/&F&6#\"\"%!\"\"/&F&6#F)F)" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "i := 'i' \+ : `seq(Eq[i])` := \{seq(Eq[i], i=1..No_Eq)\} : `seq(x[i])` := \{se q(x[i], i=1..No_x)\} :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "s olution := solve(`seq(Eq[i])`, `seq(x[i])`) : solution ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/&%\"xG6#\"\"$\"\"\"/&F&6#\"\"#F-/&F&6#\" \"%!\"\"/&F&6#F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Extracting each solution from the unorder ed solution set gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "as sign(solution) : for i to No_x do x[i] := x[i] : print(evaln(x[i] ) = x[i]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"F '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"%!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for i to No_x do x[i] := eva ln(x[i]) : od :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 395 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 396 4 "N.B." }{TEXT -1 164 " \+ The solution of a system of simultaneous equations may, on occassion, \+ become difficult due to round-off errors if a calculator is used for c omputations based on " }{TEXT 406 7 "Cramer\222" }{TEXT -1 1 "s" } {TEXT 407 2 " " }{TEXT -1 137 "rule. In general, this may take place \+ in solving systems for which the characteristic determinant of the coe fficient matrix is close to " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 32 " . Such systems are said to be " }{TEXT 397 15 "ill-conditioned" } {TEXT -1 45 ". A simple example is provided by the system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Eq[1] := x[1] + x[2] = 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Eq[2] := x[1] + 1.000 1*x[2] = 1 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Eq[1] ; E q[2] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"xG6#\"\"\"F(&F&6#\"\" #F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"xG6#\"\"\"F(*&$\"&, +\"!\"%F(&F&6#\"\"#F(F(F(" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "which, however, does not pose a pr oblem when tackled using " }{TEXT 398 5 "Maple" }{TEXT -1 74 " and any of the methods presented earlier. This is demonstrated hereunder." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 401 8 "Method 1" }{TEXT -1 9 ". Using " }{TEXT 399 7 "Cramer\222" } {TEXT -1 1 "s" }{TEXT 400 2 " " }{TEXT -1 5 "rule:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A := genmatrix([Eq[1], Eq[2]], [x[1], x[2 ]]) : A = matrix(A) ; C_A := copy(A) :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"\"F*7$F*$\"&,+\"!\"%" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "X := matrix(2, 1, [x[1], x[2]]) : X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"XG-%'matrixG6#7$7#&%\"xG6#\"\"\" 7#&F+6#\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "K := matrix(2, 1, [rhs(Eq[1]), rhs(Eq[2]) ]) : K = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'mat rixG6#7$7#\"\"%7#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) : Det(A ) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#%\"AG$\"\" \"!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(a) Finding " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`det(A, 1 <\226 K)` : = copyinto(K, A, 1, 1) : Det(A, `1 <\226 K`) = Det(matrix(`det(A, 1 \+ <\226 K)`)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'1~<| at~KG-F%6#-%'matrixG6#7$7$\"\"%\"\"\"7$F1$\"&,+\"!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`det(A, 1 <\226 K)` := det(`det(A, 1 <\226 K)`) : Det(A, `1 < \226 K`) = `det(A, 1 <\226 K)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$DetG6$%\"AG%'1~<|at~KG$\"&/+$!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x[1] := `det(A, 1 <\226 K)`/`det(A)` : 'x[1]' = x[1] ; x[1] := 'x[1]' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"$\"+++S+I!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "( b) Finding " }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := matrix(C_A) :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 95 "`det(A, 2 <\226 K)` := copyinto(K, A, 1, 2) : Det(A, `2 <\226 K`) = Det(matrix(`det(A, 2 <\226 K)`)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'2~<|at~KG-F%6#-%'matrixG6#7 $7$\"\"\"\"\"%7$F0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`det(A, 2 <\226 K)` := det(` det(A, 2 <\226 K)`) : Det(A, `2 <\226 K`) = `det(A, 2 <\226 K)` ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6$%\"AG%'2~<|at~KG!\"$" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x[2] := `det(A, 2 <\226 K)`/`det(A)` : 'x[2]' = x[2 ] ; x[2] := 'x[2]' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\" \"#$!+++++I!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 403 8 "Method 2" }{TEXT -1 12 ". Using the " } {TEXT 402 8 "linsolve" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 64 "A := genmatrix([Eq[1], Eq[2]], [x[1], x[2]]) \+ : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG 6#7$7$\"\"\"F*7$F*$\"&,+\"!\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X := linsolve(A, K) : \+ X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"XG-%'matrixG6# 7$7#$\"&/+$\"\"!7#$!&++$F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Extracting the numerical value of either unknown from the solution vector [" }{TEXT 418 1 "X" }{TEXT -1 7 "] gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i to 2 do x[i] := X[i,1] : print(evaln(x[i]) = x[i]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"$\"&/+$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#$!&++$\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 405 8 "Method 3" } {TEXT -1 12 ". Using the " }{TEXT 404 5 "solve" }{TEXT -1 10 " functio n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i to 2 do x[i] := evaln(x[i]) : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "so lution := solve(\{Eq[1], Eq[2]\}, \{x[1], x[2]\}) : solution ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"xG6#\"\"\"$\"&/+$\"\"!/&F&6#\" \"#$!&++$F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Extracting either solution from the unordered s olution set gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "assign (solution) : for i to 2 do x[i] := x[i] : print(evaln(x[i]) = x[i ]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"$\"&/+$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#$!&++$\"\"!" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i to 2 do x[i] := evaln(x[i]) : od :" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 285 "" 0 "" {TEXT 415 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Yet another example of an " }{TEXT 408 15 "ill-conditioned" }{TEXT -1 39 " system are the simultaneous equat ions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Eq[1] := 0.00001*x[ 1] + x[2] + 0.00001*x[3] = 0.00002 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Eq[2] := x[1] + 2*x[2] + \+ x[3] = 1 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Eq[3] := 0.00001*x[1] + x[2] - 0.00001*x[3] = 0.00001 :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Eq[1] ; Eq[2] ; Eq[3] ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"xG6#\"\"\"$F(!\"&&F&6#\"\"# F(*&F)F(&F&6#\"\"$F(F($F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\" xG6#\"\"\"F(*&\"\"#F(&F&6#F*F(F(&F&6#\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"xG6#\"\"\"$F(!\"&&F&6#\"\"#F(*&$F(F*F(&F&6#\"\"$ F(!\"\"F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 24 "The coefficient matrix [" }{TEXT 421 1 "A" }{TEXT -1 5 "] is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "A := genmatr ix([Eq[1], Eq[2], Eq[3]], [x[1], x[2], x[3]]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%$\"\"\"!\"&F+F*7 %F+\"\"#F+7%F*F+$!\"\"F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The characteristic determinant of \+ the coefficient matrix [" }{TEXT 422 1 "A" }{TEXT -1 16 "] has the val ue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`det(A)` := det(A) \+ : Det(A) = `det(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$DetG6#% \"AG$\"''***>!#5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The inhomogeneous vector [" }{TEXT 423 1 "K" }{TEXT -1 5 "] is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "K := matrix(3, 1, [rhs(Eq[1]), rhs(Eq[2]), rhs(Eq[3])]) : K = matrix( K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'matrixG6#7%7#$\"\"#! \"&7#\"\"\"7#$F.F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "With the default environment variable " } {TEXT 424 6 "Digits" }{TEXT -1 1 " " }{TEXT 436 1 "(" }{TEXT -1 6 "i.e . " }{XPPEDIT 18 0 "10" "6#\"#5" }{TEXT 437 1 ")" }{TEXT -1 41 ", the solution vector obtained using the " }{TEXT 420 8 "linsolve" }{TEXT -1 13 " function is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "X : = linsolve(A, K) : X = matrix(X) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"XG-%'matrixG6#7%7#$\"+'***z**\\!#57#$\"+++-+5!#97#$\"+++++]F," } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Extracting the numerical value of each unknown from the s olution vector [" }{TEXT 416 1 "X" }{TEXT -1 7 "] gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i to 3 do x[i] := X[i,1] : p rint(evaln(x[i]) = x[i]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"xG6#\"\"\"$\"+'***z**\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"xG6#\"\"#$\"+++-+5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\" \"$$\"+++++]!#5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 410 10 "Conclusion" }{TEXT 411 1 ":" }{TEXT -1 16 " The notion of " }{TEXT 409 15 "ill-conditioned" }{TEXT -1 69 " \+ systems of linear inhomogeneous equations does not apply at all if " } {TEXT 412 5 "Maple" }{TEXT -1 34 " is used for solving such systems." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 284 "" 0 "" {TEXT 414 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (20) for \"" }{TEXT 394 17 "Trace of a matrix" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 271 "" 0 "" {TEXT 393 67 "-------------------------------------------------------- -----------" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }