College Algebra 9th txtbk

478 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
SOLUTION
We are interested in finding a column matrix X that satisfies the matrix equation AX B. To
solve this equation, we multiply both sides, on the left, by A 1 , assuming it exists, to isolate
X on the left side.
AX B
A 1 (AX ) A 1 B
(A 1 A)X A 1 B
IX A 1 B
Use the left multiplication property.
Associative property
A 1 A I
IX X
X A 1 B
ZZZ CAUTION ZZZ
1. Do not mix the left multiplication property and the right multiplication property. If
AX B, then
2. Matrix division is not defined. If a, b, and x are real numbers, then the solution
of ax b can be written either as x a 1 b or as x b a. But if A, B, and X are
matrices, the solution of AX B must be written as X A 1 B. The expression
is not defined for matrices.
B
A
A 1 (AX ) BA 1
MATCHED PROBLEM 4 Given an n n matrix A and n 1 column matrices B, C, and X, solve AX C B for X.
Assume all necessary inverses exist.
Z Matrix Equations and Systems of Linear Equations
We will now show how independent systems of linear equations with the same number of
variables as equations can be solved by first converting the system into a matrix equation
of the form AX B and using X A 1 B, as obtained in Example 4.
EXAMPLE 5 Using Inverses to Solve Systems of Equations
Use matrix inverse methods to solve the system
x 1 x 2 x 3 1
2x 2 x 3 1
2x 1 3x 2 1
(4)
SOLUTION
First, we will convert the system of equations (4) into a matrix equation:
A X B
1 1 1 x 1 1
£ 0 2 1 §£ x 2 § £ 1 §
2 3 0 x 3 1
(5)
You should check that the matrix equation (5) is equivalent to the original system of equations
(4) by performing the multiplication on the left side, and then equating corresponding
elements.
If we can find the column matrix X, it will provide a solution to the system. In Example
4, we found that if AX B and A 1 exists, then X A 1 B. So our job is to find A 1