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UNIT16 - Unit 16 Matrix Equations and Inverses Denition...

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Unit 16 Matrix Equations and Inverses Definition 17.1. A set of m linear equations in n variables: a 11 x 1 + a 12 x 2 + · · · a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · a 2 n x n = b 2 . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · a mn x n = b m may be re-expressed in its matrix form as a 11 a 12 a 13 · · · a 1 n a 21 a 22 a 23 · · · a 2 n . . . . . . . . . . . . a m 1 a m 2 a m 3 · · · a mn x 1 x 2 x 3 . . . x n = b 1 b 2 b 3 . . . b n i.e. has the form Ax = b Where A is called the coefficient matrix of the system. So solving the origi- nal system is equivalent to solving Ax = b for the vector x . To this end, we introduce the inverse of a matrix. Definition 17.2. Let A be a square matrix. If there exists a matrix B with the same dimensions as A such that AB = BA = I We say that A is invertible (or nonsingular ) and that B is the inverse of A , written B = A - 1 . If A has no inverse it is said to be noninvertible (or singular ). 1
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Example 1. If A = 2 1 0 1 show that A - 1 = B = 1 / 2 1 / 2 0 1 Solution: We must show that AB = BA = I : AB = 2 1 0 1 1 / 2 1 / 2 0 1 = (2)(1 / 2) + ( 1)(0) (2)(1 / 2) + ( 1)(1) (0)(1
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