966 CHAPTER 9 Systems and Matrices ALGEBRAIC SOLUTION The 3 * 3 identity matrix is I3 = £ 1 0 0 0 1 0 0 0 1§. Using matrix multiplication, AI3 = £ -2 3 0 4 5 8 0 9 -6§ £ 1 0 0 0 1 0 0 0 1§ = £ -2 3 0 4 5 8 0 9 -6§ = A. EXAMPLE 1 Verifying the Identity Property of I 3 Let A = £ -2 3 0 4 5 8 0 9 -6§. Give the 3 * 3 identity matrix I3 and show that AI3 = A. GRAPHING CALCULATOR SOLUTION The calculator screen in Figure 28(a) shows the identity matrix for n = 3. The screen in Figure 28(b) supports the algebraic result. S Now Try Exercise 7. (a) (b) Figure 28 Multiplicative Inverses For every nonzero real number a, there is a multiplicative inverse 1 a that satisfies both of the following. a # 1 a = 1 and 1 a # a = 1 (Recall that 1 a is also written a-1.) In a similar way, if A is an n * n matrix, then its multiplicative inverse, written A−1, must satisfy both of the following. AA-1 = I n and A-1A = I n This means that only a square matrix can have a multiplicative inverse. To find the matrix A-1, we use row transformations, introduced earlier in this chapter. As an example, we find the inverse of A = c 2 1 4 -1d . Let the unknown inverse matrix be symbolized as follows. A-1 = c x z y wd By the definition of matrix inverse, AA-1 = I 2. AA-1 = c 2 1 4 -1d c x z y wd = c 1 0 0 1d NOTE Although a-1 = 1 a for any nonzero real number a, if A is a matrix, then A-1 ≠1 A . We do NOT use the symbol 1 A because 1 is a number and A is a matrix.
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