A First Course in Linear Algebra, 2017a

2.1. Matrix Arithmetic 67
Substituting in the appropriate values, this product becomes
⎡ ⎤
⎡
[ ]
b 11
a21 a 22 a 23
⎣ b 21
⎦ = [ 7 6 2 ] 1
⎣ 3
b 31 2
⎤
⎦ = 1 × 7 + 3 × 6 + 2 × 2 = 29
Hence, (AB) 21 = 29.
You should take a moment to find a few other entries of AB. You can multiply the matrices to check
that your answers are correct. The product AB is given by
⎡ ⎤
13 13
AB = ⎣ 29 32 ⎦
0 0
♠
2.1.5 Properties of Matrix Multiplication
As pointed out above, it is sometimes possible to multiply matrices in one order but not in the other order.
However, even if both AB and BA are defined, they may not be equal.
Example 2.24: Matrix Multiplication is Not Commutative
[ 1 2
3 4
Compare the products AB and BA,formatricesA =
]
,B =
[ 0 1
1 0
]
Solution. First, notice that A and B are both of size 2×2. Therefore, both products AB and BA are defined.
The first product, AB is
[ ][ ] [ ]
1 2 0 1 2 1
AB =
=
3 4 1 0 4 3
The second product, BA is [ 0 1
1 0
Therefore, AB ≠ BA.
][ 1 2
3 4
]
=
[ 3 4
1 2
]
♠
This example illustrates that you cannot assume AB = BA even when multiplication is defined in both
orders. If for some matrices A and B it is true that AB = BA, thenwesaythatA and B commute. Thisis
one important property of matrix multiplication.
The following are other important properties of matrix multiplication. Notice that these properties hold
only when the size of matrices are such that the products are defined.