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Lecture Notes (19)

# Lecture Notes (19) - MA 3280 Lecture 19 Determinants Monday...

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MA 3280 Lecture 19 - Determinants Monday, March 24, 2014. Objectives: Material from 8.2 and 8.3: Order 1, 2, and 3 determinants. At the beginning of the semester, and when we talked about Fnding inverse matrices, we used something that is called a determinant . ±or example, given a 2 × 2 matrix (1) A = b a 11 a 12 a 21 a 22 B , we saw that (2) A - 1 = 1 a 11 a 22 - a 12 a 21 b a 22 - a 12 - a 21 a 11 B That funny thing on the bottom is the determinant of A . We’ll use vertical line brackets to denote the determinant, like | A | . A 2 × 2 determinant, therefore, is deFned to be (3) v v v v a 11 a 12 a 21 a 22 v v v v = a 11 a 22 - a 12 a 21 . As an example, (4) v v v v 3 4 7 2 v v v v = (3)(2) - (4)(7) = 6 - 28 = - 22 . Cramer’s Rule. The little trick I showed you to solve a 2 × 2 system of equations can be reformulated in terms of determinants. It turns out that the solution for a system like (5) 2 x 1 + 3 x 2 = 1 5 x 1 + 4 x 2 = 6 has solutions as follows. (6) x 1 = v v v v 1 3 6 4 v v v v v v v v 2 3 5 4 v v v v = 4 - 18 8 - 15 = - 14 - 7 = 2 , and (7) x 2 = v v v v 2 1 5 6 v v v v v v v v 2 3 5 4 v v v v = 12 - 5 8 - 15 = 7 - 7 = - 1 . Do you see the pattern? The determinant on the bottom has the coe²cients on the left side of the equations,

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