solq8 - Answer: matrix AB is not necessarily symmetric; b)...

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Solutions to quiz # 8 (October 28) 1. We look for the projection in the form x 1 1 0 + y 0 1 1 , for some numbers x and y such that 1 1 1 - x 1 1 0 - y 0 1 1 · 1 1 0 = 0 and 1 1 1 - x 1 1 0 - y 0 1 1 · 0 1 1 = 0 , that is, 2 - 2 x - y = 0 and 2 - x - 2 y = 0 . Solving the system of linear equations, we get x y 2 1 | 2 1 2 | 2 -→ ± 1 2 | 2 2 1 | 2 ² -→ ± 1 2 | 2 0 - 3 | - 2 ² -→ ± 1 0 | 2 / 3 0 1 | 2 / 3 ² , from which we get x = y = 2 / 3 and the projection is 2 3 1 1 0 + 2 3 0 1 1 = 2 / 3 4 / 3 2 / 3 . Answer: The projection is 2 / 3 4 / 3 2 / 3 . 2. Since A and B symmetric, we have A T = A and B T = B . a) We have ( AB ) T = B T A T = BA . Since matrices AB and BA are not necessarily equal, we get the following
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Unformatted text preview: Answer: matrix AB is not necessarily symmetric; b) We have ( ABA ) T = A T B T A T = ABA . Answer: matrix ABA is symmetric; c) We have ( A-1 ) T = ( A T )-1 = A-1 . Answer: matrix A-1 is symmetric; d) We have ( A-1 BA ) T = A T B T ( A-1 ) T = AB ( A T )-1 = ABA-1 . Since matrices A-1 BA and ABA-1 are not necessarily equal, we get the following Answer: matrix A-1 BA is not necessarily symmetric. 1...
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