MATRICES AND MATRIX OPERATIONS: Unit 6
Dr. Wlodzislaw Kostecki
The Papua New Guinea University of Technology (PNGUT)
Department of Electrical and Communication Engineering
Lae, Morobe Province
Papua New Guinea
Copyright © 2000 by Wlodzislaw Kostecki
All rights reserved
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(6) Multiplication of a multi-row multi-column matrix and a column matrix
OBJECTIVES :
• To introduce this operation as a basis for solving systems of linear algebraic inhomogeneous equations using Cramer’ s rule.
• To provide examples of multiplication of a multi-row multi-column matrix and a column matrix.
> restart : with(linalg, multiply) :
According to the multiplication conformability rule, the product of a rectangular matrix of order
(
×
)
and an
(
×
)
column matrix
is possible only if the number of columns in the
(
×
)
matrix is equal to the number of rows in the
(
×
)
column matrix
, i.e.
. The product matrix is an
(
×
)
column matrix
.
For example, consider a
(
×
)
matrix [
A
] given as
> 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) ;
![A = matrix([[a[11], a[12], a[13], a[14]], [a[21], a...](images10/Matrices-Unit614.gif)
and a
(
×
)
column matrix
[
CM
] given as
> CM := matrix(4, 1, [cm[11], cm[21], cm[31], cm[41]]) : CM = matrix(CM) ;
![CM = matrix([[cm[11]], [cm[21]], [cm[31]], [cm[41]]...](images10/Matrices-Unit617.gif)
The product [
A
] [
CM
] is the following
(
× 1)
column matrix:
> `A CM` := multiply(A, CM) : A*CM = matrix(`A CM`) ;
![A*CM = matrix([[a[11]*cm[11]+a[12]*cm[21]+a[13]*cm[...](images10/Matrices-Unit619.gif)
This matrix multiplication may be displayed in "like-in-a-book" form, namely
> matrix(A) * matrix(CM) = matrix(`A CM`) ;
![matrix([[a[11], a[12], a[13], a[14]], [a[21], a[22]...](images10/Matrices-Unit620.gif)
Numerical example for this type of matrix multiplication
Let a
(
×
)
matrix [
A
] and a
(
×
)
column matrix [
CM
] be given as
> A := matrix(3, 4, [1, 2, 1, 0, 0, 1, 1, 3, 1, 2, 1, 4]) : CM := matrix(4, 1, [2, 1, 0, -1]) :
> A = matrix(A) ; CM = matrix(CM) ;
![A = matrix([[1, 2, 1, 0], [0, 1, 1, 3], [1, 2, 1, 4...](images10/Matrices-Unit625.gif)
![CM = matrix([[2], [1], [0], [-1]])](images10/Matrices-Unit626.gif)
The product [
A
] [
CM
] is the following
(
× 1)
column matrix:
> `A CM` := multiply(A, CM) : A*CM = matrix(`A CM`) ;
![A*CM = matrix([[4], [-2], [0]])](images10/Matrices-Unit628.gif)
or, in "like-in-a-book" form,
> matrix(A) * matrix(CM) = matrix(`A CM`) ;
![matrix([[1, 2, 1, 0], [0, 1, 1, 3], [1, 2, 1, 4]])*...](images10/Matrices-Unit629.gif)
* * *
N.B. This specific type of matrix multiplication finds application in solving systems of linear algebraic inhomogeneous equations.
[ For solving systems of linear algebraic inhomogeneous equations, refer to Unit (19). ]
* * *
Proceed to Unit (7) for " Special types of matrices ".
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