Multiplying Matrices
Definition:
A matrix is a rectangular array of elements arranged in rows and columns.
⎡ a11
⎢
[ A ] = ⎢ a i1
⎢ a m1
⎣
a1 j
a ij
a mj
a m1
a1 n ⎤
⎥
a in ⎥
a mn ⎥⎦
amj
a1n
ain
amn
R
O
W
S
amn
COLUMNS
m = number of rows
n = number of columns
aij =element in row i and column j
m x n = dimension of matrix
BASIC LAWS:
Commutative:
[ AB] ≠ [ BA]
Distributive:
[ A]([ B ] + [C ]) = [ AB ] + [ AC ]
Associative:
[ A]([ BC ]) = ([ AB ])[C ]
([ A] + [ B ])[C ] = [ AC ] + [ BC ]
Things to remember:
1. In general, [AB] ≠ [BA].
2. If [AB] = [AC], then it is not true in general that [B] = [C].
3. If [AB] = 0, then it is not true in general that [A] = 0 or [B] = 0.
4. Powers of matrices: [A]n = [A]……[A]
n
MULTIPLICATION:
If [A] is an m x n matrix, and if [B] is an n x p matrix, then the product
[AB] is the m x p matrix that we will call [E].
2x3
3x2
[A ]
⎡ a11
⎢a
⎣ 21
a12
a22
x
a13 ⎤
a23 ⎥⎦
x
[B ]
⎡ b11
⎢b
⎢ 21
⎢⎣b31
The Math Center
2x2
= [E ]
b12 ⎤
a (b ) + a12 (b21 ) + a13 (b31 ) a11 (b12 ) + a12 (b22 ) + a13 (b32 ) ⎤
b22 ⎥⎥ = ⎡⎢ 11 11
a 21 (b11 ) + a 22 (b21 ) + a 23 (b31 ) a 21 (b12 ) + a 22 (b22 ) + a 23 (b32 ) ⎥⎦
⎣
b32 ⎥⎦
■
Valle Verde
■
Tutorial Support Services
■
EPCC
1
EXAMPLE 1:
1 4 5⎤
[A] = ⎡⎢
⎥
⎣8 2 9⎦
⎡3
[B ] = ⎢⎢ 6
⎢⎣ 4
8⎤
2 ⎥⎥
5 ⎥⎦
FIND [AB].
[A] is a 2 x 3 matrix, and [B] is a 3 x 2 matrix. The product [AB] is 2 x 2 matrix.
⎡1
⎢8
⎣
4
2
⎡3
5⎤
X ⎢⎢ 6
⎥
9⎦
⎢⎣ 4
8⎤
⎡ 1(3 ) + 4 ( 6 ) + 5 ( 4 )
2 ⎥⎥ = ⎢
⎣8 (3) + 2 (6 ) + 9 ( 4 )
5 ⎥⎦
⎡ 47
[ AB ] = ⎢
⎣ 72
1(8 ) + 4 ( 2 ) + 5 (5 ) ⎤
8 ( 8 ) + 2 ( 2 ) + 9 ( 5 ) ⎥⎦
41 ⎤
113 ⎥⎦
EXAMPLE 2:
1 4 5⎤
[A] = ⎡⎢
⎥
⎣8 2 9⎦
⎡3
[B ] = ⎢⎢ 6
⎢⎣ 4
8⎤
2 ⎥⎥
5 ⎥⎦
FIND [BA].
[B] is a 3 x 2 matrix, and [A] is a 2 x 3 matrix. The product [BA] is 3 x 3 matrix.
⎡3
⎢6
⎢
⎢⎣ 4
8⎤
2 ⎥⎥ X
5 ⎥⎦
⎡1
⎢8
⎣
4
2
⎡ 3 (1 ) + 8 ( 8 )
5⎤
= ⎢⎢ 6 (1 ) + 2 ( 8 )
9 ⎥⎦
⎢⎣ 4 (1 ) + 5 ( 8 )
⎡ 67
[ BA ] = ⎢⎢ 22
⎢⎣ 44
3(4 ) + 8(2 )
6(4) + 2(2)
4(4) + 5(2)
3(5 ) + 8 (9 ) ⎤
6 ( 5 ) + 2 ( 9 ) ⎥⎥
4 ( 5 ) + 5 ( 9 ) ⎥⎦
87 ⎤
48 ⎥⎥
65 ⎥⎦
28
28
26
*Notice that the product [AB] does not equal to the product [BA].*
The Math Center
■
Valle Verde
■
Tutorial Support Services
■
EPCC
2