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lecture_15 - MA 265 LECTURE NOTES MONDAY FEBRUARY 11...

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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, FEBRUARY 11 Cramer’s Rule Let A be an n × n matrix. We have seen that if det( A ) 6 = 0, then A is nonsingular. In fact, its inverse is the n × n matrix A- 1 = 1 det( A ) adj( A ) = A ij det( A ) T = A ji det( A ) . We give a brief example. Consider a 2 × 2 matrix: A = a b c d . We have the adjoint adj A = d- c- b a T = d- b- c a = ⇒ A- 1 = d ad- bc- b ad- bc- c ad- bc a ad- bc . Linear Systems. Let’s return to a system of linear equations: 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 n 1 x 1 + a n 2 x 2 + ··· + a nn x n = b n where the number of equations is the same as the number of unknowns. Recall that we can express this system as a product of matrices: A = a 11 a 12 ... a 1 n a 21 a 22 ... a 2 n . . . . . . . . . . . . a n 1 a n 2 ... a nn , x = x 1 x 2 . . . x n and b = b 1 b 2 . . . b n = ⇒ A x = b . Hence if det( A ) 6 = 0, then A is nonsingular, so that the unique solution so this system is x = A- 1 b . Cramer’s Rule. Now that we have an explicit formula for A- 1 , we use this to work out an explicit formula for x . Substituting in the expressions above: x = A- 1 b = 1 det( A ) · adj( A ) · b = 1 det( A ) A 11 A 21 ··· A n 1 A 12 A 22 ··· A n 2 ....
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lecture_15 - MA 265 LECTURE NOTES MONDAY FEBRUARY 11...

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