Page 126 - Matrix Analysis & Applied Linear Algebra

120 Chapter 3 Matrix Algebra

Solving a nonsingular system Ax = b by first computing A −1 and then
2
3
forming the product x = A −1 b requires n + n multiplications/divisions and
2
3
n −n additions/subtractions. Recall from §1.5 that Gaussian elimination with
3
back substitution requires only about n /3 multiplications/divisions and about
3
n /3 additions/subtractions. In other words, using A −1 to solve a nonsingular
system Ax = b requires about three times the effort as does Gaussian elimina-
tion with back substitution.
To put things in perspective, consider standard matrix multiplication be-
tween two n × n matrices. It is not difficult to verify that n 3 multiplications
2
3
and n −n additions are required. Remarkably, it takes almost exactly as much
effort to perform one matrix multiplication as to perform one matrix inversion.
This fact always seems to be counter to a novice’s intuition—it “feels” like ma-
trix inversion should be a more difficult task than matrix multiplication, but this
is not the case.
The remainder of this section is devoted to a discussion of some of the
important properties of matrix inversion. We begin with the four basic facts
listed below.

Properties of Matrix Inversion
For nonsingular matrices A and B, the following properties hold.
−1 −1
• A = A. (3.7.13)
• The product AB is also nonsingular. (3.7.14)
• (AB) −1 = B −1 A −1 (the reverse order law for inversion). (3.7.15)
−1 T T −1 −1 ∗ ∗ −1
• A = A and A =(A ) . (3.7.16)

Proof. Property (3.7.13) follows directly from the definition of inversion. To
prove (3.7.14) and (3.7.15), let X = B −1 A −1 and verify that (AB)X = I by
writing

(AB)X =(AB)B −1 A −1 = A(BB −1 )A −1 = A(I)A −1 = AA −1 = I.

According to the discussion in Example 3.7.2, we are now guaranteed that
X(AB)= I, and we need not bother to verify it. To prove property (3.7.16), let
−1 T T
X = A and verify that A X = I. Make use of the reverse order law for
transposition to write
T −1 T T

−1
T
T
A X = A A = A A = I = I.
−1 T
Therefore, A T = X = A −1 . The proof of the conjugate transpose case
is similar.