[− 123
011
− 102
][ 2 − 4 − 1
− 111
1 − 2 − 1
]
=
[− 100
0 − 10
00 − 1
]
=−
[ 100
010
001
]
=−I
whereIis called the 3× 3 unit matrix. It follows that if
A=
[
− 123
011
− 102
]
and
B=
[− 241
1 − 1 − 1
− 121
]
(note the overall change of sign) then
AB=I
You can also check that
BA=I
We say in these circumstances thatBis theinverseofA– and then of courseAis the
inverse ofB. We will see how to find the inverse matrix in Section 13.6.
Problem 13.7
Evaluate
[
1 − 1
00
][
12
12
]
The product is the 2×2 zero matrix
[
00
00
]
This is an example ofAB= 0 even though neitherAnorBis zero.
Exercises on 13.3
- If
[
x+y 3
− 1 x−y
]
=
[
2 X+Y
Y−X 1
]
findx,y,X,Y.