College Algebra 9th txtbk

SECTION 7–4 Solving Systems of Linear Equations Using Matrix Inverse Methods 471
ZZZ EXPLORE-DISCUSS 1
(A) Pick any 2 2 matrix you like, and multiply it by the following matrix in both
possible orders.
c 1 0
0 1 d
(B) Repeat (A) for any 3 3 matrix you like, but multiply by the matrix
What can you conclude?
1 0 0
£ 0 1 0§
0 0 1
Z DEFINITION 1 Identity Matrix
The identity matrix for multiplication for the set of all square matrices of order
n is the square matrix of order n, denoted by I, with 1’s along the principal diagonal
(from upper left corner to lower right corner) and 0’s elsewhere.
In Explore-Discuss 1, we saw that
c 1 0
0 1 d
and
1 0 0
£ 0 1 0§
0 0 1
are the identity matrices for square matrices of order 2 and 3, respectively.
We will show in Exercises 7-4 that if M is any square matrix of order n and I is the
identity matrix of order n, then
IM MI M
Note: If M is an m n matrix that is not square (m n), then it is still possible to multiply
M on the left and on the right by an identity matrix, but not with the same-size identity
matrix. To avoid the complications involved with associating two different identity
matrices with each nonsquare matrix, we will restrict our attention in this section to square
matrices.
Z Finding the Inverse of a Square Matrix
In the set of real numbers, we know that for each real number a, except 0, there exists a
real number a 1 such that
a 1 a 1
The number a 1 is called the inverse of the number a relative to multiplication, or the multiplicative
inverse of a. For example, 2 1 is the multiplicative inverse of 2, since 2 1 (2) 1.
We will use this idea to define the inverse of a square matrix.