{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 Input" 2 19 "" 0 1 255 0 0 1 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 "" 0 1 0 0 0 0 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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Helvetica" 0 1 0 0 128 1 0 1 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 0 1 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 0 1 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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 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 "" 0 1 0 0 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "Courier" 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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 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 0 1 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 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 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 0 0 0 0 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 0 0 0 0 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 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 308 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE " " -1 319 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "T imes" 0 1 0 0 0 0 0 0 2 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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 324 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "" 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 1 0 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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 331 "" 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 "" 0 1 0 0 0 0 0 1 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 "Courier" 0 1 0 0 0 0 0 1 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 "" 0 1 0 0 0 0 0 1 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 341 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 343 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 347 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 348 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 350 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 352 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 353 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 354 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 355 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 356 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 357 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 358 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 361 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 363 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 364 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 365 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 366 "" 0 1 0 0 0 0 0 0 2 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 369 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 370 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 371 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 372 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 373 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 374 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 375 "Tim es" 0 1 0 0 0 0 0 0 2 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 378 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 379 "" 0 1 0 0 0 0 1 0 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 383 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 384 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 385 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 386 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 387 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 388 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 389 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 390 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 391 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 392 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 393 "" 0 1 0 0 0 0 0 0 2 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 "" 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 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 397 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 398 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 399 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 400 "Tim es" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 401 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 402 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 403 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 404 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 405 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 406 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 407 "Courier" 0 1 0 0 0 0 0 1 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 "Helvetica" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 410 "Helvetica" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 411 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 412 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 413 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 414 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 415 "" 0 1 128 0 0 1 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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 418 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 419 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 420 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 421 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 422 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 423 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 424 "Tim es" 0 1 0 0 0 0 0 0 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 "Times" 0 1 0 0 0 0 0 0 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 "Times" 0 1 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 431 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 432 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 433 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 434 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 435 "Times" 0 1 0 0 0 0 0 0 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 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 439 "Times" 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 "Times" 0 1 0 0 0 0 0 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 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 445 "Tim es" 0 1 0 0 0 0 0 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 "Times" 0 1 0 0 0 0 0 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 "Tim es" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 453 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 454 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 455 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 456 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 457 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 458 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 459 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 460 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 461 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 462 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 463 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 464 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 465 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 466 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 467 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 468 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 469 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 470 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 471 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 472 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 473 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 474 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 475 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 476 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 477 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 478 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 479 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 480 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 481 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE " " -1 482 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 483 " " 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 484 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 485 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 486 "Helvetica" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 487 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 488 "" 0 1 0 0 64 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 489 "Helvetica" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 490 " " 0 1 111 0 7 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 491 "" 0 1 111 0 7 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 492 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 493 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 494 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 495 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 496 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 497 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 498 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 499 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 500 " Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 501 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 502 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 503 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 504 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 505 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 506 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 507 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 508 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 509 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 510 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 511 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 512 " Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 513 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 514 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 515 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 516 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 517 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 518 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 519 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 520 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 521 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 522 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 523 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 524 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 525 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 526 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 527 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 528 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 529 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 530 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 531 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 532 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 533 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 534 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 535 "Times " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 536 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 537 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 538 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 539 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 540 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 541 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 542 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 543 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 544 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 545 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 546 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 547 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 548 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 549 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 550 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 551 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 552 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 553 "Tim es" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 554 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 555 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 556 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 557 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 558 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 559 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 560 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 561 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 562 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 563 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 564 "Times " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 565 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 566 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 567 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 568 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 569 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 570 "T imes" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 571 "Times" 0 1 0 0 0 0 1 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 572 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 573 "Times" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 574 "Helvetica" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 575 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 576 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 577 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 578 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 579 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 580 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 581 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 582 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 583 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 584 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 585 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 586 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 587 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 588 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 589 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 590 "Times" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 591 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 592 "Times" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 593 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 594 "Tim es" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 595 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 596 "Times" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 597 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 598 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 599 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 600 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 601 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 602 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 603 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 604 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 605 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 606 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 607 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 608 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 609 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 610 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 611 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 612 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 613 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 614 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 615 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 616 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 128 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE " " -1 -1 "Courier" 1 12 0 128 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Arial Narro w" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 265 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 266 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 267 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 268 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 269 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 270 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 271 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 272 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 273 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 274 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 275 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 276 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 277 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 278 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 279 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 280 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 281 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 282 1 {CSTYLE "" -1 -1 "Arial Nar row" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 283 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 284 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 285 1 {CSTYLE " " -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 273 "" 0 "" {TEXT 486 38 "MATRICES AND MATRIX OPE RATIONS: Unit 4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 274 "" 0 "" {TEXT 488 23 "Dr. Wlodzislaw Kostecki" }}{PARA 275 "" 0 "" {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }} {PARA 276 "" 0 "" {TEXT -1 54 "Department of Electrical and Communicat ion Engineering" }}{PARA 277 "" 0 "" {TEXT -1 20 "Lae, Morobe Province " }}{PARA 278 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 " " {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT 487 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 280 "" 0 "" {TEXT 489 19 "All rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT 490 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 265 3 "(4)" }{TEXT 410 1 " " }{TEXT 409 26 "Multiplication of \+ matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 574 10 "OBJECTIVES" }{TEXT 575 1 ":" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 576 1 "\225" } {TEXT -1 66 " To define the operation of matrix multiplication and st ate the " }{TEXT 609 34 "multiplication conformability rule" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 611 1 "\225" }{TEXT -1 32 " \+ To introduce the concepts of " }{TEXT 612 14 "pre-multiplier" }{TEXT -1 7 " and " }{TEXT 613 15 "post-multiplier" }{TEXT -1 11 " matrice s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 607 1 "\225" }{TEXT -1 63 " To pr ovide alternative methods of matrix multiplication with " }{TEXT 608 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 610 1 "\225 " }{TEXT -1 54 " To specify the laws obeyed by matrix multiplication. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 615 1 "\225" }{TEXT -1 59 " To intr oduce the concept of, and provide an example of, " }{TEXT 614 9 "comm uting" }{TEXT -1 11 " matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 616 1 "\225" }{TEXT -1 87 " To state and illustrate the various propertie s of operation of matrix multiplication." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart : \+ with(linalg, multiply) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The matrices to be multiplied " } {TEXT 19 4 "must" }{TEXT -1 12 " obey the " }{TEXT 258 34 "multiplic ation conformability rule" }{TEXT -1 16 ", which states:" }}{PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "\"Two matr ices can only be multiplied together if the number of " }{TEXT 259 7 "columns" }{TEXT -1 58 " of the first (left-hand one) is equal to the number of " }{TEXT 260 4 "rows" }{TEXT -1 17 " of the second.\"" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Such matrices are called " }{TEXT 346 11 "conformable" }{TEXT -1 6 " or " }{TEXT 347 10 "compatible" }{TEXT -1 41 " for multiplic ation. If the matrices do " }{TEXT 348 3 "not" }{TEXT -1 37 " satisfy \+ this rule, their product is " }{TEXT 344 3 "not" }{TEXT -1 15 " define d (does " }{TEXT 345 3 "not" }{TEXT -1 8 " exist)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "If two ma trices to be multiplied are [" }{TEXT 261 1 "A" }{TEXT -1 12 "] of ord er " }{TEXT 418 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 419 3 " \+ \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 420 1 ")" }{TEXT -1 17 " w ith elements " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 7 " and [" }{TEXT 262 1 "B" }{TEXT -1 12 "] of order " }{TEXT 421 1 "( " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 422 3 " \327 " }{XPPEDIT 18 0 "p " "6#%\"pG" }{TEXT 423 1 ")" }{TEXT -1 17 " with elements " } {XPPEDIT 18 0 "b[ij]" "6#&%\"bG6#%#ijG" }{TEXT 578 1 "," }{TEXT -1 28 " then the matrix product [" }{TEXT 263 1 "A" }{TEXT -1 3 "] [" } {TEXT 264 1 "B" }{TEXT -1 18 "] is the matrix [" }{TEXT 577 1 "C" } {TEXT -1 12 "] of order " }{TEXT 424 1 "(" }{XPPEDIT 18 0 "m" "6#%\"m G" }{TEXT 425 3 " \327 " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT 426 1 ") " }{TEXT -1 17 " with elements " }{XPPEDIT 18 0 "c[ij]" "6#&%\"cG6#% #ijG" }{TEXT -1 47 " that are found from the following definition." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The element of the " }{XPPEDIT 18 0 "i" "6#%\"iG" } {TEXT -1 12 "th row and " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 14 " th column of [" }{TEXT 599 1 "C" }{TEXT -1 62 "] is obtained by summin g the products of each element of the " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 15 "th row of the " }{TEXT 597 5 "first" }{TEXT -1 49 " ma trix with the corresponding elements of the " }{XPPEDIT 18 0 "j" "6#% \"jG" }{TEXT -1 18 "th column of the " }{TEXT 598 6 "second" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Thus, the general element of the product \+ matrix [" }{TEXT 600 1 "C" }{TEXT -1 13 "] is given by" }}}{EXCHG {PARA 285 "" 0 "" {XPPEDIT 18 0 "c[ij]=a[i1]*b[1j]+a[i2]*b[2j]" "6#/&% \"cG6#%#ijG,&*&&%\"aG6#%#i1G\"\"\"&%\"bG6#*&F.F.%\"jGF.F.F.*&&F+6#%#i2 GF.&F06#*&\"\"#F.F3F.F.F." }{TEXT 589 8 " + ... +" }{TEXT -1 1 " " } {XPPEDIT 18 0 "a[ip]*b[pj]=Sum(a[ik]*b[kj], k=1..p)" "6#/*&&%\"aG6#%#i pG\"\"\"&%\"bG6#%#pjGF)-%$SumG6$*&&F&6#%#ikGF)&F+6#%#kjGF)/%\"kG;F)%\" pG" }{TEXT -1 9 " " }{TEXT 590 1 "i" }{TEXT 591 10 " = 1,..., \+ " }{TEXT 592 1 "m" }{TEXT 593 5 " ; " }{TEXT 594 1 "j" }{TEXT 595 10 " = 1,..., " }{TEXT 596 1 "p" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The scalar " }{XPPEDIT 18 0 "c[ij]" "6#&%\"cG6#%#ijG" }{TEXT -1 17 " is called the " }{TEXT 601 13 "inner product" }{TEXT -1 10 " of the " }{XPPEDIT 18 0 "i" "6 #%\"iG" }{TEXT -1 11 "th row of [" }{TEXT 602 1 "A" }{TEXT -1 8 "] wit h " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 14 "th column of [" } {TEXT 603 1 "B" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 604 4 "N.B." }{TEXT -1 125 " Disti nction should be made between the above inner product of a row of one \+ matrix with a column of another matrix and the " }{TEXT 605 13 "inner product" }{TEXT -1 8 " of a " }{TEXT 606 32 "sequence of vectors an d matrices" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The first factor in the matrix pr oduct is called the " }{TEXT 579 14 "pre-multiplier" }{TEXT -1 32 " \+ and the second is called the " }{TEXT 580 15 "post-multiplier" } {TEXT -1 20 ". In the product [" }{TEXT 581 1 "A" }{TEXT -1 3 "] [" }{TEXT 582 1 "B" }{TEXT -1 12 "], matrix [" }{TEXT 583 1 "A" }{TEXT -1 3 "] " }{TEXT 587 14 "pre-multiplies" }{TEXT -1 3 " [" }{TEXT 584 1 "B" }{TEXT -1 28 "], or, alternately, matrix [" }{TEXT 585 1 "A " }{TEXT -1 6 "] is " }{TEXT 588 15 "post-multiplied" }{TEXT -1 6 " \+ by [" }{TEXT 586 1 "B" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, cons ider a " }{TEXT 568 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 569 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 570 1 ")" }{TEXT -1 10 " \+ matrix [" }{TEXT 567 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "A := matrix(3, 4, [a[11], a[12], a[13], \+ a[14], a[21], a[22], a[23], a[24], a[31], a[32], a[33], a[34]]) : A \+ = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7 &&%\"aG6#\"#6&F+6#\"#7&F+6#\"#8&F+6#\"#97&&F+6#\"#@&F+6#\"#A&F+6#\"#B& F+6#\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and a " } {TEXT 427 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 428 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 429 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 266 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "B := matrix(4, 2, [b[11], b[12], b[21], b[22], b[3 1], b[32], b[41], b[42]]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7&7$&%\"bG6#\"#6&F+6#\"#77$&F+6#\"#@& F+6#\"#A7$&F+6#\"#J&F+6#\"#K7$&F+6#\"#T&F+6#\"#U" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The produc t [" }{TEXT 282 1 "A" }{TEXT -1 3 "] [" }{TEXT 283 1 "B" }{TEXT -1 9 "] is a " }{TEXT 430 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 431 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 432 1 ")" }{TEXT -1 82 " matrix, which may be obtained using either of the following alterna tive methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 411 8 "Method 1" }{TEXT -1 12 ". Using the " } {TEXT 412 8 "multiply" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "`AB` := multiply(A, B) : A*B = matrix(`AB`) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"BGF&-%'matrixG6# 7%7$,**&&%\"aG6#\"#6F&&%\"bGF1F&F&*&&F06#\"#7F&&F46#\"#@F&F&*&&F06#\"# 8F&&F46#\"#JF&F&*&&F06#\"#9F&&F46#\"#TF&F&,**&F/F&&F4F7F&F&*&F6F&&F46# \"#AF&F&*&F=F&&F46#\"#KF&F&*&FDF&&F46#\"#UF&F&7$,**&&F0F:F&F3F&F&*&&F0 FOF&F9F&F&*&&F06#\"#BF&F@F&F&*&&F06#\"#CF&FGF&F&,**&FfnF&FLF&F&*&FhnF& FNF&F&*&FjnF&FRF&F&*&F^oF&FVF&F&7$,**&&F0FAF&F3F&F&*&&F0FSF&F9F&F&*&&F 06#\"#LF&F@F&F&*&&F06#\"#MF&FGF&F&,**&FioF&FLF&F&*&F[pF&FNF&F&*&F]pF&F RF&F&*&FapF&FVF&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 413 8 "Method 2" }{TEXT -1 12 ". Using the " } {TEXT 414 5 "evalm" }{TEXT -1 27 " function and the operator " }{TEXT 415 2 "&*" }{TEXT -1 10 " (called " }{TEXT 417 9 "amperstar" }{TEXT -1 5 " in " }{TEXT 416 5 "Maple" }{TEXT -1 55 ") that indicates non-c ommutative matrix multiplication:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "`AB` := evalm(A &* B) : A*B = matrix(`AB`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"BGF&-%'matrixG6#7%7$, **&&%\"aG6#\"#6F&&%\"bGF1F&F&*&&F06#\"#7F&&F46#\"#@F&F&*&&F06#\"#8F&&F 46#\"#JF&F&*&&F06#\"#9F&&F46#\"#TF&F&,**&F/F&&F4F7F&F&*&F6F&&F46#\"#AF &F&*&F=F&&F46#\"#KF&F&*&FDF&&F46#\"#UF&F&7$,**&&F0F:F&F3F&F&*&&F0FOF&F 9F&F&*&&F06#\"#BF&F@F&F&*&&F06#\"#CF&FGF&F&,**&FfnF&FLF&F&*&FhnF&FNF&F &*&FjnF&FRF&F&*&F^oF&FVF&F&7$,**&&F0FAF&F3F&F&*&&F0FSF&F9F&F&*&&F06#\" #LF&F@F&F&*&&F06#\"#MF&FGF&F&,**&FioF&FLF&F&*&F[pF&FNF&F&*&F]pF&FRF&F& *&FapF&FVF&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "This matrix multiplication may be display ed in \"like-in-a-book\" form, namely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "matrix(A) * matrix(B) = matrix(`AB`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&-%'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#\"#M\"\"\"-F&6#7&7$&%\"bGF,&FWF/7$&FWF9&FWF<7$&FWFF&FWFI7$ &FW6#\"#T&FW6#\"#UFQ-F&6#7%7$,**&F*FQFVFQFQ*&F.FQFZFQFQ*&F1FQFgnFQFQ*& F4FQFjnFQFQ,**&F*FQFXFQFQ*&F.FQFenFQFQ*&F1FQFhnFQFQ*&F4FQF]oFQFQ7$,**& F8FQFVFQFQ*&F;FQFZFQFQ*&F>FQFgnFQFQ*&FAFQFjnFQFQ,**&F8FQFXFQFQ*&F;FQFe nFQFQ*&F>FQFhnFQFQ*&FAFQF]oFQFQ7$,**&FEFQFVFQFQ*&FHFQFZFQFQ*&FKFQFgnFQ FQ*&FNFQFjnFQFQ,**&FEFQFXFQFQ*&FHFQFenFQFQ*&FKFQFhnFQFQ*&FNFQF]oFQFQ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 42 "Numerical example of matrix multiplication" }}{PARA 2 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let a " }{TEXT 433 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 434 3 " \327 " } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 435 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 277 1 "A" }{TEXT -1 9 "] and a " }{TEXT 436 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 437 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 438 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 278 1 "B" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A := \+ matrix(3, 4, [1, 2, 1, 0, 0, 1, 1, 3, 1, 2, 1, 4]) : B := matrix(4, \+ 2, [2, 1, 1, 2, 0, 2, -1, 1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7&\"\"\"\"\"#F*\"\"!7&F,F*F*\"\"$7& F*F+F*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7&7$ \"\"#\"\"\"7$F+F*7$\"\"!F*7$!\"\"F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" }{TEXT 279 1 "A" }{TEXT -1 3 "] [" }{TEXT 280 1 "B" }{TEXT -1 21 "] is the f ollowing " }{TEXT 439 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 440 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 441 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`AB` := multip ly(A, B) : A*B = matrix(`AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&%\"AG\"\"\"%\"BGF&-%'matrixG6#7%7$\"\"%\"\"(7$!\"#F.7$\"\"!\"#6" }}} {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 38 "matrix(A) * matrix(B) = matrix(`AB`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7%7&\"\"\"\"\"#F*\"\"!7&F,F*F*\" \"$7&F*F+F*\"\"%F*-F&6#7&7$F+F*7$F*F+7$F,F+7$!\"\"F*F*-F&6#7%7$F0\"\"( 7$!\"#F=7$F,\"#6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 286 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 4 "N.B." }{TEXT -1 42 " I n general, matrix multiplication does " }{TEXT 19 3 "not" }{TEXT -1 12 " obey the " }{TEXT 267 15 "commutative law" }{TEXT -1 7 ", i.e. " }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 1 "[" }{TEXT 289 1 "A" }{TEXT -1 3 "] [" }{TEXT 290 1 "B" }{TEXT -1 3 "] " }{TEXT 19 15 "is not equ al to" }{TEXT -1 3 " [" }{TEXT 291 1 "B" }{TEXT -1 3 "] [" }{TEXT 292 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "since the product [" }{TEXT 293 1 "B" }{TEXT -1 3 "] [" }{TEXT 294 1 "A" }{TEXT -1 7 "] may " }{TEXT 295 3 "not" }{TEXT -1 72 " satisfy the multiplication conformability r ule, which means that it is " }{TEXT 296 3 "not" }{TEXT -1 40 " define d. Even if it is defined, it may " }{TEXT 299 3 "not" }{TEXT -1 27 " b e equal to the product [" }{TEXT 297 1 "A" }{TEXT -1 3 "] [" }{TEXT 298 1 "B" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "For example, consider two " } {TEXT 442 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 443 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 444 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 300 1 "A" }{TEXT -1 7 "] and [" }{TEXT 301 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "A := mat rix(2, 2, [1, -1, 0, 4]) : B := matrix(2, 2, [1, -2, 3, 6]) : A = \+ matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"A G-%'matrixG6#7$7$\"\"\"!\"\"7$\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"\"\"!\"#7$\"\"$\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The p roduct [" }{TEXT 302 1 "A" }{TEXT -1 3 "] [" }{TEXT 303 1 "B" }{TEXT -1 21 "] is the following " }{TEXT 445 1 "(" }{XPPEDIT 18 0 "2" "6# \"\"#" }{TEXT 446 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 447 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AB` := multiply(A, B) : `A B` = matrix(`AB`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%$A~BG-%'matrixG6#7$7$!\"#!\")7$\"#7\"#C" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product [" }{TEXT 304 1 "B" }{TEXT -1 3 "] [" }{TEXT 305 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 448 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 449 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 450 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`BA` := multiply(B, A) : `B A` = matrix(`BA `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$B~AG-%'matrixG6#7$7$\"\"\"! \"*7$\"\"$\"#@" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "It is obvious that the two product matric es of (a) and (b) are " }{TEXT 306 3 "not" }{TEXT -1 7 " equal." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 307 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 4 "N.B." }{TEXT -1 31 " If two matrices sat isfy the " }{TEXT 288 11 "commutative" }{TEXT -1 29 " law of multipl ication, i.e." }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 269 1 "A" }{TEXT -1 3 "] [" }{TEXT 270 1 "B" }{TEXT -1 1 "]" }{TEXT 451 3 " = " }{TEXT -1 1 "[" }{TEXT 271 1 "B" }{TEXT -1 3 "] [" }{TEXT 272 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "they are said to be " }{TEXT 379 9 "comm uting" }{TEXT -1 11 " matrices." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 27 "For example, consider two " }{TEXT 452 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 311 3 " \327 \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 312 1 ")" }{TEXT 453 12 " matri ces [" }{TEXT 309 1 "A" }{TEXT 313 7 "] and [" }{TEXT 310 1 "B" } {TEXT 314 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "A := matrix(2, 2, [6, 8, 4, 6]) : B := matrix(2, 2, [15, 20, 10 , 15]) : A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"'\"\")7$\"\"%F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7$\"#:\"#?7$\"#5F*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The product [" }{TEXT 273 1 "A" }{TEXT -1 3 "] [" }{TEXT 274 1 "B" }{TEXT -1 20 "] is the following " }{TEXT 315 1 " " }{TEXT 454 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 316 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 317 1 ")" }{TEXT 455 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AB` := multiply(A, B) : `A B` = matrix(`AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A~BG-%' matrixG6#7$7$\"$q\"\"$S#7$\"$?\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product [" } {TEXT 275 1 "B" }{TEXT -1 3 "] [" }{TEXT 276 1 "A" }{TEXT -1 20 "] is the following " }{TEXT 318 1 " " }{TEXT 456 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 319 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 320 1 ")" }{TEXT 457 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`BA` := multiply(B, A) : `B A` = matrix(`BA`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%$B~AG-%'matrixG6#7$7$\"$q\"\"$S#7$\" $?\"F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Both product matrices are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 285 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 380 4 "N.B." }{TEXT -1 20 " The matrix product" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 1 "[" }{TEXT 381 1 "A" }{TEXT -1 3 "] [" } {TEXT 382 1 "B" }{TEXT -1 1 "]" }{TEXT 458 3 " = " }{TEXT -1 1 "[" } {TEXT 383 1 "0" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "does " }{TEXT 384 3 "not" } {TEXT -1 33 " necessarily imply either that [" }{TEXT 385 1 "A" } {TEXT -1 1 "]" }{TEXT 459 3 " = " }{TEXT -1 1 "[" }{TEXT 386 1 "0" } {TEXT -1 13 "] or that [" }{TEXT 387 1 "B" }{TEXT -1 1 "]" }{TEXT 460 3 " = " }{TEXT -1 1 "[" }{TEXT 388 1 "0" }{TEXT -1 2 "]." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 389 27 "For example, consider two " }{TEXT 461 1 "(" }{XPPEDIT 18 0 " 3" "6#\"\"$" }{TEXT 397 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 398 1 ")" }{TEXT 462 12 " matrices [" }{TEXT 390 1 "A" }{TEXT 392 7 "] and [" }{TEXT 391 1 "B" }{TEXT 393 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "A := matrix(3, 3, [1, -1, 1, -3, 2 , -1, -2, 1, 0]) : B := matrix(3, 3, [1, 2, 3, 2, 4, 6, 1, 2, 3]) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = m atrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\" \"\"!\"\"F*7%!\"$\"\"#F+7%!\"#F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%F+\"\"%\"\"'F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "T he product [" }{TEXT 394 1 "A" }{TEXT -1 3 "] [" }{TEXT 395 1 "B" } {TEXT -1 8 "] is a " }{TEXT 396 1 " " }{TEXT 463 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 399 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 400 1 ")" }{TEXT 464 2 " " }{TEXT 571 4 "zero" }{TEXT 572 8 " \+ matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`AB` := multiply( A, B) : `A B` = matrix(`AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% $A~BG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Notice that the product [" }{TEXT 404 1 "B" }{TEXT -1 3 "] [" }{TEXT 405 1 "A" }{TEXT -1 6 " ] is " }{TEXT 406 3 "not" }{TEXT -1 4 " a " }{TEXT 573 4 "zero" } {TEXT -1 34 " matrix, but it is the following " }{TEXT 401 1 " " } {TEXT 465 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 402 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 403 1 ")" }{TEXT 466 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`BA` := multiply(B, A) : `B A` = matrix(`BA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$B~AG-%' matrixG6#7%7%!#6\"\"'!\"\"7%!#A\"#7!\"#F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "which is yet anot her example for the statement that matrix multiplication does " } {TEXT 19 3 "not" }{TEXT -1 12 " obey the " }{TEXT 408 15 "commutativ e law" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 272 "" 0 "" {TEXT 407 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 4 "N.B." }{TEXT -1 28 " Matrix multiplication is " }{TEXT 321 11 "associative" } {TEXT -1 10 ", so that" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 1 "[" } {TEXT 330 1 "A" }{TEXT -1 1 "]" }{TEXT 467 2 " (" }{TEXT -1 1 "[" } {TEXT 331 1 "B" }{TEXT -1 3 "] [" }{TEXT 332 1 "C" }{TEXT -1 1 "]" } {TEXT 468 5 ") = (" }{TEXT -1 1 "[" }{TEXT 333 1 "A" }{TEXT -1 3 "] [ " }{TEXT 334 1 "B" }{TEXT -1 1 "]" }{TEXT 469 2 ") " }{TEXT -1 1 "[" } {TEXT 335 1 "C" }{TEXT -1 1 "]" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 470 1 "(" }{TEXT -1 1 "[" }{TEXT 324 1 "A" }{TEXT -1 3 "] [" }{TEXT 325 1 "B" }{TEXT -1 1 "]" }{TEXT 471 2 ") " }{TEXT -1 1 "[" }{TEXT 326 1 "C " }{TEXT -1 1 "]" }{TEXT 472 3 " = " }{TEXT -1 1 "[" }{TEXT 327 1 "A" }{TEXT -1 1 "]" }{TEXT 473 2 " (" }{TEXT -1 1 "[" }{TEXT 328 1 "B" } {TEXT -1 3 "] [" }{TEXT 329 1 "C" }{TEXT -1 1 "]" }{TEXT 474 1 ")" }} {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 336 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 322 4 "N.B." }{TEXT -1 28 " Matrix multiplication is " } {TEXT 323 37 "distributive with respect to addition" }{TEXT -1 10 ", \+ so that" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 1 "[" }{TEXT 337 1 "A" } {TEXT -1 1 "]" }{TEXT 475 2 " (" }{TEXT -1 1 "[" }{TEXT 338 1 "B" } {TEXT -1 1 "]" }{TEXT 476 3 " + " }{TEXT -1 1 "[" }{TEXT 339 1 "C" } {TEXT -1 1 "]" }{TEXT 477 4 ") = " }{TEXT -1 1 "[" }{TEXT 340 1 "A" } {TEXT -1 3 "] [" }{TEXT 341 1 "B" }{TEXT -1 1 "]" }{TEXT 478 3 " + " } {TEXT -1 1 "[" }{TEXT 342 1 "A" }{TEXT -1 3 "] [" }{TEXT 343 1 "C" } {TEXT -1 1 "]" }}}{EXCHG {PARA 282 "" 0 "" {TEXT 497 1 "(" }{TEXT -1 1 "[" }{TEXT 493 1 "B" }{TEXT -1 1 "]" }{TEXT 498 3 " + " }{TEXT -1 1 "[" }{TEXT 494 1 "C" }{TEXT -1 1 "]" }{TEXT 499 2 ") " }{TEXT -1 1 "[ " }{TEXT 501 1 "A" }{TEXT -1 1 "]" }{TEXT 502 3 " = " }{TEXT -1 1 "[" }{TEXT 495 1 "B" }{TEXT -1 1 "]" }{TEXT 500 1 " " }{TEXT -1 1 "[" } {TEXT 503 1 "A" }{TEXT -1 2 "] " }{TEXT 504 2 "+ " }{TEXT -1 1 "[" } {TEXT 496 1 "C" }{TEXT -1 3 "] [" }{TEXT 505 1 "A" }{TEXT -1 1 "]" }}} {EXCHG {PARA 283 "" 0 "" {TEXT 506 1 "(" }{TEXT -1 1 "[" }{TEXT 518 1 "A" }{TEXT -1 2 "] " }{TEXT 507 1 "+" }{TEXT -1 2 " [" }{TEXT 519 1 "B " }{TEXT -1 1 "]" }{TEXT 508 1 ")" }{TEXT -1 1 " " }{TEXT 509 1 "(" } {TEXT -1 1 "[" }{TEXT 520 1 "M" }{TEXT -1 2 "] " }{TEXT 510 1 "+" } {TEXT -1 2 " [" }{TEXT 521 1 "N" }{TEXT -1 1 "]" }{TEXT 511 4 ") = " } {TEXT -1 1 "[" }{TEXT 522 1 "A" }{TEXT -1 2 "] " }{TEXT 512 1 "(" } {TEXT -1 1 "[" }{TEXT 523 1 "M" }{TEXT -1 2 "] " }{TEXT 513 1 "+" } {TEXT -1 2 " [" }{TEXT 524 1 "N" }{TEXT -1 1 "]" }{TEXT 514 4 ") + " } {TEXT -1 1 "[" }{TEXT 525 1 "B" }{TEXT -1 2 "] " }{TEXT 515 1 "(" } {TEXT -1 1 "[" }{TEXT 526 1 "M" }{TEXT -1 2 "] " }{TEXT 516 1 "+" } {TEXT -1 2 " [" }{TEXT 527 1 "N" }{TEXT -1 1 "]" }{TEXT 517 1 ")" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For example, consider " }{TEXT 529 1 "(" }{XPPEDIT 18 0 "3" "6 #\"\"$" }{TEXT 530 3 " \327 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 531 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 528 1 "A" }{TEXT -1 7 "] and \+ [" }{TEXT 532 1 "B" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "A:=matrix(3, 4, [1, -2, 1, 0, 6, 1, 1, 3, -1, 2 , 1, 4]) : B:=matrix(3, 4, [3, 2, -2, 1, 0, -3, 2, 1, 3, 2, 5, -1]) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = m atrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7&\" \"\"!\"#F*\"\"!7&\"\"'F*F*\"\"$7&!\"\"\"\"#F*\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7&\"\"$\"\"#!\"#\"\"\"7&\"\"!!\"$ F+F-7&F*F+\"\"&!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "and " }{TEXT 534 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 535 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 536 1 ")" }{TEXT -1 12 " matrices [" }{TEXT 533 1 "M" }{TEXT -1 7 "] and [" }{TEXT 537 1 "N" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "M := matrix(4, 2, [1, 2, -1, 3, 0, \+ -2, -1, 7]) : N := matrix(4, 2, [4, -6, 5, -2, 3, 2, -4, 1]) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "M = matrix(M) ; N = matrix (N) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"MG-%'matrixG6#7&7$\"\"\" \"\"#7$!\"\"\"\"$7$\"\"!!\"#7$F-\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"NG-%'matrixG6#7&7$\"\"%!\"'7$\"\"&!\"#7$\"\"$\"\"#7$!\"%\"\"\" " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "(a) The product " }{TEXT 538 1 "(" }{TEXT -1 1 "[" } {TEXT 544 1 "A" }{TEXT -1 2 "] " }{TEXT 539 1 "+" }{TEXT -1 2 " [" } {TEXT 545 1 "B" }{TEXT -1 1 "]" }{TEXT 540 1 ")" }{TEXT -1 1 " " } {TEXT 541 1 "(" }{TEXT -1 1 "[" }{TEXT 546 1 "M" }{TEXT -1 2 "] " } {TEXT 542 1 "+" }{TEXT -1 2 " [" }{TEXT 547 1 "N" }{TEXT -1 1 "]" } {TEXT 543 1 ")" }{TEXT -1 20 " is the following " }{TEXT 548 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 549 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 550 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`(A+B) (M+N)` := evalm((A+B) &* (M+N)) : '( A+B) (M+N)' = matrix(`(A+B) (M+N)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-,&%\"AG\"\"\"%\"BGF'6#,&%\"MGF'%\"NGF'-%'matrixG6#7%7$\"#7!\")7$ \"#6\"\"'7$\"#H\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "(b) The sum of the products [" } {TEXT 557 1 "A" }{TEXT -1 2 "] " }{TEXT 551 1 "(" }{TEXT -1 1 "[" } {TEXT 558 1 "M" }{TEXT -1 2 "] " }{TEXT 552 1 "+" }{TEXT -1 2 " [" } {TEXT 559 1 "N" }{TEXT -1 1 "]" }{TEXT 553 4 ") + " }{TEXT -1 1 "[" } {TEXT 560 1 "B" }{TEXT -1 2 "] " }{TEXT 554 1 "(" }{TEXT -1 1 "[" } {TEXT 561 1 "M" }{TEXT -1 2 "] " }{TEXT 555 1 "+" }{TEXT -1 2 " [" } {TEXT 562 1 "N" }{TEXT -1 1 "]" }{TEXT 556 1 ")" }{TEXT -1 20 " is th e following " }{TEXT 563 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 564 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 565 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`A(M+N) \+ + B(M+N)` := evalm(A&*(M+N) + B&*(M+N)) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "'A(M+N) + B(M+N)' = matrix(`A(M+N) + B(M+N)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"AG6#,&%\"MG\"\"\"%\"NGF*F*-%\"B GF'F*-%'matrixG6#7%7$\"#7!\")7$\"#6\"\"'7$\"#H\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The resul tant matrices of (a) and (b) are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 284 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 349 4 "N.B." }{TEXT -1 7 " The " }{TEXT 357 16 "cancellation law" } {TEXT -1 5 " is " }{TEXT 350 3 "not" }{TEXT -1 41 " valid for matrix \+ multiplication, i.e. if" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 1 "[" } {TEXT 351 1 "A" }{TEXT -1 3 "] [" }{TEXT 352 1 "B" }{TEXT -1 1 "]" } {TEXT 479 3 " = " }{TEXT -1 1 "[" }{TEXT 353 1 "A" }{TEXT -1 3 "] [" } {TEXT 354 1 "C" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "then" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 1 "[" }{TEXT 355 1 "B" }{TEXT -1 3 "] " }{TEXT 19 15 "i s not equal to" }{TEXT -1 3 " [" }{TEXT 356 1 "C" }{TEXT -1 1 "]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 358 29 "For example, consider three " }{TEXT 480 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 363 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 364 1 ")" }{TEXT 481 12 " matrices [" }{TEXT 359 1 "A" }{TEXT 361 4 "], [" }{TEXT 360 1 "B" }{TEXT 362 8 "], and [" }{TEXT 365 1 "C " }{TEXT 366 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "A := matrix(3, 3, [4, 2, 0, 2, 1, 0, -2, -1, 1]) : B := matrix( 3, 3, [2, 3, 1, 2, -2, -2, -1, 2, 1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "C := matrix(3, 3, [3, 1, -3, 0, 2, 6, -1, 2, 1]) : \+ A = matrix(A) ; B = matrix(B) ; C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"%\"\"#\"\"!7%F+\"\"\"F,7%! \"#!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\" \"#\"\"$\"\"\"7%F*!\"#F.7%!\"\"F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%\"CG-%'matrixG6#7%7%\"\"$\"\"\"!\"$7%\"\"!\"\"#\"\"'7%!\"\"F/F+" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The product [" }{TEXT 367 1 "A" }{TEXT -1 3 "] [" }{TEXT 368 1 "B" }{TEXT -1 19 "] is the following" }{TEXT 369 2 " " }{TEXT 482 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 370 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 371 1 ")" }{TEXT 483 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`AB` := multiply(A, B) : A*B = matrix(`AB`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\" \"%\"BGF&-%'matrixG6#7%7%\"#7\"\")\"\"!7%\"\"'\"\"%F/7%!\"(!\"#F&" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The product [" }{TEXT 372 1 "A" }{TEXT -1 3 "] [" }{TEXT 376 1 "C" }{TEXT -1 19 "] is the following" }{TEXT 373 2 " " }{TEXT 484 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 374 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 375 1 ")" }{TEXT 485 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`AC` := multiply(A, C) : A*C = matrix(`AC`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\" \"%\"CGF&-%'matrixG6#7%7%\"#7\"\")\"\"!7%\"\"'\"\"%F/7%!\"(!\"#F&" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Although the product matrices of (a) and (b) are equal, the mat rices [" }{TEXT 377 1 "B" }{TEXT -1 7 "] and [" }{TEXT 378 1 "C" } {TEXT -1 16 "] are different." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 284 "" 0 "" {TEXT 566 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (5) for \"" }{TEXT 492 41 "Multiplication of row and column matr ices" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 270 "" 0 "" {TEXT 491 67 "---- ---------------------------------------------------------------" }}}} {MARK "163 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }