Example 10-3. Note that only in special cases is matrix multiplication com-
mutative. One can say in general AB =BA. Consider the matrix product of the 2 × 2
matrices given by A=
(
1 2
0 1)and B=
(
0 3
1 1).One finds
AB =(
1 2
0 1)(
0 3
1 1)
=(
2 5
1 1)and
BA =(
0 3
1 1)(
1 2
0 1)
=(
0 3
1 3)which shows that in general matrix multiplication is not commutative.
In addition, if the matrix product of Aand Bproduces AB = [0], this does not
mean A= [0] or B= [0]. For example, if A=
(
1 1
− 1 − 1)and B=
(
−1 1
1 − 1), then
one can show
AB =(
1 1
− 1 − 1)(
−1 1
1 − 1)
=(
0 0
0 0)Special Square Matrices
There are many special matrices which have interesting properties. The following
are some definitions of special square matrices which arise in applied mathematics,
engineering, physics and the sciences.
The identity matrix
The n×nidentity matrix can be expressed I= (δij)n×nwhere δij is the Kronecker
delta and defined
δij ={1 if i=j
0 if i=j
This matrix is characterized by having all 1’s along the main diagonal and zero’s
everywhere else. An example of a 3 × 3 identity matrix is given by
I=
1 0 0
0 1 0
0 0 1
The identity matrix has the property
AI =IA =A (10.6)