970 CHAPTER 9 Systems and Matrices EXAMPLE 3 Identifying a Matrix with No Inverse Find A-1, if possible, given that A = c 2 1 -4 -2d . ALGEBRAIC SOLUTION Using row transformations to change the first column of the augmented matrix c 2 1 -4 -2 2 1 0 0 1d results in the following matrices. c 1 1 -2 -2 3 1 2 0 0 1d 1 2 R1 c1 0 -2 0 3 1 2 - 1 2 0 1d -R1 + R2 At this point, the matrix should be changed so that the second row, second element will be 1. Because that element is now 0, there is no way to complete the desired transformation, so A-1 does not exist for this matrix A. Just as there is no multiplicative inverse for the real number 0, not every matrix has a multiplicative inverse. Matrix A is an example of such a matrix. GRAPHING CALCULATOR SOLUTION If the inverse of a matrix does not exist, the matrix is called singular, as shown in Figure 29 for matrix [A]. This occurs when the determinant of the matrix is 0. Figure 29 S Now Try Exercise 21. Solution of Systems Using Inverse Matrices Matrix inverses can be used to solve square linear systems of equations. (A square system has the same number of equations as variables.) For example, consider the following linear system of three equations with three variables. a11x + a12 y + a13z = b1 a21x + a22 y + a23z = b2 a31x + a32 y + a33z = b3 The definition of matrix multiplication can be used to rewrite the system using matrices. £a11 a21 a31 a12 a22 a32 a13 a23 a33§ # £ x y z§ = £ b1 b2 b3§ (1) (To see this, multiply the matrices on the left.) If A = £ a11 a21 a31 a12 a22 a32 a13 a23 a33§, X = £ x y z§, and B = £ b1 b2 b3§, then the system given in (1) becomes AX = B. If A-1 exists, then each side of AX = B can be multiplied on the left as shown on the next page.
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