solq10 - the inverse of an orthogonal matrix is an...

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Solutions to quiz # 10 (November 11) 1. We compute det 1 1 2 3 2 4 2 10 1 1 2 5 1 3 2 8 = det 1 1 2 3 0 2 2 4 0 0 0 2 0 2 0 5 = det 1 1 2 3 0 2 2 4 0 2 0 5 0 0 0 2 = det 1 1 2 3 0 2 2 4 0 0 2 1 0 0 0 2 = 8 . Answer: det 1 1 2 3 2 4 2 10 1 1 2 5 1 3 2 8 = 8. 2. Since A and B are orthogonal matrices, we have A T = A - 1 and B T = B - 1 . Then ( AB ) T = B T A T = B - 1 A - 1 = ( AB ) - 1 , so AB is an orthogonal matrix; ( AB - 1 ) T = ( AB T ) T = ( B T ) T A T = BA T = BA - 1 = ( AB - 1 ) - 1 , so AB - 1 is an orthogonal matrix; if A = b 1 0 0 1 B and B = A then both A and B are orthogonal matrices but A + B is not; (2 A ) T = 2 A T = 2 A - 1 n = 1 2 A - 1 = (2 A ) - 1 , so 2 A is not an orthogonal matrix; ( A T B ) T = B T ( A T ) T = B - 1 A = ( A - 1 B ) - 1 = ( A T B ) - 1 , so A T B is an orthogonal matrix. More generally, since the product of orthogonal matrices is an orthogonal matrix and
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Unformatted text preview: the inverse of an orthogonal matrix is an orthogonal matrix, a product of any number of orthogonal matrices and their inverses is an orthogonal matrix. Answer: matrices AB, AB-1 and A T B are necessarily orthogonal, matrix A + B does not have to be orthogonal and matrix 2 A is never orthogonal. 1...
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