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Stitz-Zeager_College_Algebra_e-book

522 86 systems of equations and matrices partial

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Unformatted text preview: 1 (5) into the original equation ? 3x = 5 3 3−1 (5) ? =5 ? 3 · 3−1 (5) = 5 Associative property of multiplication ? 1·5 = 5 Inverse property 5=5 Multiplicative Identity Thinking back to Theorem 8.5, we know that matrix multiplication enjoys both an associative property and a multiplicative identity. What’s missing from the mix is a multiplicative inverse for the coefficient matrix A. Assuming we can find such a beast, we can mimic our solution (and check) to 3x = 5 as follows 1 Every nonzero real number a has a multiplicative inverse, denoted a−1 , such that a−1 · a = a · a−1 = 1. 494 Systems of Equations and Matrices Solving AX = B Checking our answer ? AX A−1 (AX ) A−1 A X I2 X X = = = = = AX = B B A−1 B A−1 B A−1 B A−1 B A A−1 B ? =B ? AA−1 B = B ? I2 B = B B=B The matrix A−1 is read ‘A-inverse’ and we will define it formally later in the section. At this stage, we have no idea if such a matrix A−1 exists, but that won’t deter us from trying to find it.2 We want A−1 to satisfy two equations, A−1 A = I2 and AA−1 = I2 , maki...
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