Stitz-Zeager_College_Algebra_e-book

# Certainly x2 and x2 1 but are there any other factors

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Unformatted text preview: ng A−1 necessarily a 2 × 2 matrix.3 Hence, we assume A−1 has the form A−1 = x1 x2 x3 x4 for real numbers x1 , x2 , x3 and x4 . For reasons which will become clear later, we focus our attention on the equation AA−1 = I2 . We have AA−1 = I2 x1 x2 x3 x4 = 10 01 2x1 − 3x3 2x2 − 3x4 3x1 + 4x3 3x2 + 4x4 = 10 01 2 −3 3 4 This gives rise to two more systems of equations 2x1 − 3x3 = 1 3x1 + 4x3 = 0 2x2 − 3x4 = 0 3x2 + 4 x4 = 1 At this point, it may seem absurd to continue with this venture. After all, the intent was to solve one system of equations, and in doing so, we have produced two more to solve. Remember, the objective of this discussion is to develop a general method which, when used in the correct scenarios, allows us to do far more than just solve a system of equations. If we set about to solve these systems using augmented matrices using the techniques in Section 8.2, we see that not only do both systems have the same coeﬃcient matrix, this coeﬃcient matrix is none other than...
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