340spring06finalsol

340spring06finalsol - TOTAL SCORE (135 points possible):...

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Unformatted text preview: TOTAL SCORE (135 points possible): MATH 340; FINAL EXAM, May 12, 2006 (R.A.Brualdi) Discussion Section ( circle one ): Mon 8:50 Mon 12:05 Wed 8:50 Wed 12:05 NAME: 1. [27 points] A and B are real matrices of order 4 with determinants 3, and 7 respectively. Answer the following questions: 1. det(- A ) = (- 1) 4 3 = 3 2. det A T = 3 3. det B- 1 = 7- 1 = 1 / 7 4. det( AB ) = 3 7 = 21 5. If 1 + i and 2- 3 i are complex eigenvalues of A , what are its other two eigenvalues? 1- i and 2 + 3 i . 6. The row space of A equals: R 4 7. The product of the eigenvalues of A T equals det A T = 3: 8. The product of the eigenvalues of B- 1 equals (- 1) 4 det B- 1 = 1 / 7: 9. The dot product of the first column vector of A with the second row vector of A- 1 equals: 0 (since A- 1 A = I 4 ) 1 2. [14 points] Let A = bracketleftBigg 8 3- 1 4 bracketrightBigg . Determine an invertible matrix Q that diagonalizes A : Q- 1 AQ = D and the diagonal matrix D . The characteristic plynomial of A equals 2- 12 + 35 = ( - 5)( - 7) . Hence the eigenvalues of A are 5 , 7. We need to find an eigenvector of A for each of these eigenvalues. 5 I 2- A = bracketleftBigg- 3- 3 1 1 bracketrightBigg ....
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340spring06finalsol - TOTAL SCORE (135 points possible):...

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