Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
We have a soluon, so
⎡
⎤
0.5 0.5 0
A = ⎣ 0.2 −0.4 0.2 ⎦ .
−0.3 0.1 0.2
Mulply AA −1 to verify that it is indeed the inverse of A. .
In general, given a matrix A, to find A −1 we need to form the augmented matrix
[ A I
]
and put it into reduced row echelon form and interpret the result. In the case
of a 2 × 2 matrix, though, there is a shortcut. We give the shortcut in terms of a
theorem. 18
.
Ṫheorem 7
The Inverse of a 2×2 Matrix
Let
[ a b
A =
c d
.
]
.
A is inverble if and only if ad − bc ≠ 0.
If ad − bc ≠ 0, then
A −1 =
1
ad − bc
[ d −b
−c a
]
.
We can’t divide by 0, so if ad − bc = 0, we don’t have an inverse. Recall Example
54, where
[ ] 1 2
A = .
2 4
Here, ad − bc = 1(4) − 2(2) = 0, which is why A didn’t have an inverse.
Although this idea is simple, we should pracce it.
. Example 56 .Use Theorem 7 to find the inverse of
if it exists.
A =
[ 3
] 2
−1 9
18 We don’t prove this theorem here, but it really isn’t hard to do. Put the matrix
[ a b 1
] 0
c d 0 1
108
into reduced row echelon form and you’ll discover the result of the theorem. Alternavely, mulply A by
what we propose is the inverse and see that we indeed get I.