hw7 - Winter 2011 • • Math 67 Linear Algebra Homework 7...

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Unformatted text preview: Winter 2011 • • Math 67 Linear Algebra Homework 7 Problem A. (this is Artin 1.3.1) Compute the following determinants: (1) 1 i 2−i 3 (2) 11 1 −1 (3) 201 010 102 (4) 1 5 8 0 0 2 6 9 0 0 3 7 0 0 0 4 (5) 1 2 4 2 4 3 1 0 1 5 0 0 3 0 0 0 Problem B. (this is Artin 1.3.8) Let A be an n-by-n matrix. What is det(−A)? (You do not need to hand this in.) Problem C. Let P and Q be matrices of size n-by-n and m-by-m. Form a new matrix of the form P0 0Q This matrix has dimensions (n + m)-by-(n + m). Prove that det P0 = (det P )(det Q) 0Q using the definition of the determinant. Do any example where n = m = 2, but do not hand this part in. Problem D. Compute the eigenvalues of the matrix in A(5), above. (Use a calculator to compute the zeros of the polynomial det(A − λI ). There is one online at http://www.convertalot.com/quartic_root_calculator.html) 1 2 Problem F. Compute the inverse of 111 2 3 4 4 9 16 using Cramer’s rule. Problem G. Suppose that A is an upper triangular matrix. Prove that A−1 is also upper triangular using Cramer’s rule. ...
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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.

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hw7 - Winter 2011 • • Math 67 Linear Algebra Homework 7...

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