Page 150 - 35Linear Algebra

150 Matrices

(b) We saw in Chapter 1 that the operator B = u× (cross product
with a vector) is a linear operator. It can therefore be written as
a matrix (given an ordered basis such as the standard basis). How
is it that composing such linear operators is non-associative even
though matrix multiplication is associative?

7.5 Inverse Matrix

Definition A square matrix M is invertible (or nonsingular) if there
exists a matrix M −1 such that

M −1 M = I = MM −1 .

If M has no inverse, we say M is singular or non-invertible.

Inverse of a 2 × 2 Matrix Let M and N be the matrices:

a b d −b
M = , N =
c d −c a
Multiplying these matrices gives:

ad − bc 0
MN = = (ad − bc)I .
0 ad − bc

d −b
1
Then M −1 = , so long as ad − bc 6= 0.
ad−bc −c a

7.5.1 Three Properties of the Inverse

1. If A is a square matrix and B is the inverse of A, then A is the inverse
of B, since AB = I = BA. So we have the identity

−1 −1
(A ) = A.

−1
−1
−1
−1
2. Notice that B A AB = B IB = I = ABB A −1 so
−1
(AB) −1 = B A −1

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