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e483h1s_07 - 1 ENSC 483 HW#1 Solution–M Saif ENSC 483...

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ENSC 483 HW#1 Solution–M. Saif 1 ENSC 483 - Modern Control Systems Simon Fraser University School of Engineering Science 1. Use Laplace’s Expansion to evaluate the determinants of the following matrices: A = 3 0 1 2 4 3 1 1 2 ; B = 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 Solution: det( A ) = (3) × det 4 3 1 2 + (1) × det 2 4 1 1 = 13 det( B ) = (1) × det 3 4 5 4 5 6 5 6 7 - (2) × det 2 4 5 3 5 6 4 6 7 +(3) × det 2 3 5 3 4 6 4 5 7 - (4) × det 2 3 4 3 4 5 4 5 6 = 0 2. Find the inverse of matrices in problem 1. Solution: By definition A - 1 = Adj ( A ) det( A ) , then Adj ( A ) = 5 - 1 - 2 1 5 - 3 - 4 - 7 12 T = A - 1 = 1 13 5 1 - 4 - 1 5 - 7 - 2 - 3 12 The matrix B does not have an inverse since it is singular. 3. (a) You are told that matrices A , and B are symmetric and that their product is symmetric. What can you conclude from this? (b) A matrix is called idempotent if A 2 = A . Consider the case of 2 × 2 idempotent matrices and say as much as you can about them, and based on your arguments give an example of a 2 × 2 idempotent matrix. Solution: (a). AB = ( AB ) T = B T A T = BA = A and B commute. (b). Let A = a b c d = A 2 = a 2 + bc ( a + d ) b ( a + d ) c d 2 + bc = a b c d . Thus we conclude that a + d = 1 , a 2 + bc = a , and d 2 + bc = d . So let a = 4 , and d = - 3 which implies bc = - 12 , and thus pick b = 3 , and c = - 4 , this gives A = 4 3 - 4 - 3 which you can verify to be an idempotent matrix. Similarly, you can come up with other examples.
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ENSC 483 HW#1 Solution–M. Saif 2 4. Show that the set of all real n × n matrices with usual operation of matrix addition and the usual operation of multiplication of matrices by scalars constitutes a vector space over the reals (i.e., < n × n , < ).
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