week09 - Math 43 Fall 2007 B Dodson Week 9 Monday Finish Suggested Hw8 start on Hw9 material 1 Determinants 2 Properties of Dets 3 Eigenvalues and

# week09 - Math 43 Fall 2007 B Dodson Week 9 Monday Finish...

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Math 43, Fall 2007 B. Dodson Week 9: Monday: Finish Suggested Hw8, start on Hw9 material 1. Determinants 2. Properties of Dets 3. Eigenvalues and Eigenvectors (2-by-2 in 4.1; then n-by-n in 4.3) ——– We compute det 2 1 5 4 2 3 9 5 1 using the (first) row expansion (by minors): det ( A ) = 2 fl fl fl fl 2 3 5 1 fl fl fl fl - 1 fl fl fl fl 4 3 9 1 fl fl fl fl + 5 fl fl fl fl 4 2 9 5 fl fl fl fl = 2(2 - 15) - (4 - 27) + 5(20 - 18) = 2( - 13) - ( - 23) + 5(2) = - 26 + 33 = 7 . ———– Problem Reduce A = 2 1 3 5 3 0 1 2 4 1 4 3 5 2 5 3 to an upper triagular matrix and use the reduction to find det ( A ) . Solution: A - 1 1 2 3 3 0 1 2 4 1 4 3 5 2 5 3 ( r 1 r 1 - r 2 ) - 1 1 2 3 0 3 7 11 0 5 12 15 0 7 15 18 ( r 2 r 2 + 3 r 1 , r 3 r 3 + 4 r 1 , r 4 r 4 + 5 r 1 )
2 - 1 1 2 3 0 3 7 11 0 5 12 15 0 1 1 - 4 ( r 4 r 4 - 2 r 2 ) ——– - 1 1 2 3 0 1 1 - 4 0 5 12 15 0 3 7 11 ( r 2 r 4 ) - 1 1 2 3 0 1 1 - 4 0 0 7 35 0 0 4 23 ( r 3 r 3 - 5 r 2 , r 4 r 4 - 3 r 2 ) - 1 1 2 3 0 1 1 - 4 0 0 - 1 - 11 0 0 4 23 ( r 3 r 3 - 2 r 4 ) - 1 1 2 3 0 1 1 - 4 0 0 - 1 - 11 0 0 0 - 21 = U ( r 4 r 4 + 4 r 3 ). Now we have det U = ( - 1)(1)( - 1)( - 21) = - 21 , and det A = ( - 1) det U = 21 , since all of the 3rd elementary operations change the determinant by a factor of (1) – so no change - there are no 2nd EROs and exactly one 1st ERO, with each 1st ERO changing the determinant by a factor of ( - 1) . Notice that, for the problem of finding the value of the determinant, we do not need a reduced row echelon or 1’s on the diagonal, and stop at a triangular form.
3 3. Eigenvalues and Eigenvectors ————— A vector v = 0 in R n is an eigenvector with eigenvalue λ of an n -by- n matrix A if Av = λv. We re-write the vector equation as ( A - λI n ) v = 0 , which is a homogeneous system with coef matrix ( A - λI ) , and we want λ so that the system has a non-trivial solution.