e303k - A and B are nxn matrices. a) If S = { v 1 ,v 2 ...v...

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MAS 3105 Feb 18, 2003 Quiz 3 Prof. S. Hudson 1) Answer True or False: If A is row equivalent to B , then det A = det B . If A is nonsingular, then A T is nonsingular. If A is m × n then N ( A ) is a subspace of R n . For all 5x5 matrices B and C, det (BC) = det (CB). If A is nonsingular then its columns are linearly independent. Answer: FTTTT 2) Short answer! Label your answers clearly: a) Find det ( A T A - 1 ), b) Find det (adj A ), c) Find det(3 A ). Where: A = 1 3 5 0 2 3 0 0 1 Answer: After noting that det A = 2, you can ignore A , and use theorems. You don’t have to compute A - 1 etc. For a), use 2 × 1 2 = 1. For b), start with A adj A = det A I , and take a det of both sides; det A det(adj A ) = (det A ) 3 det I so that det(adj A ) = 8 / 2 = 4. It is also OK to compute adj A , which isn’t too hard in this example. For c), det 3 A = 3 3 det A = 27 × 2 = 54. 3) Choose ONE of these to prove. Assume
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Unformatted text preview: A and B are nxn matrices. a) If S = { v 1 ,v 2 ...v n } , then span( S ) is a subspace of V . b) For all nxn matrices, det( AB ) = det( A ) det( B ). c) If A has two identical rows, then det( A ) = 0. (Use induction. Possible +2 bonus for choosing this one!). Answer: Parts a) and b) are in the text, though the proof of b) on page 112 has only minimal explanation. 3c) This is HW 2.1-10. Dont include specic examples. Do include a basis step ( n = 2) and an induction step ( n > 2). You can then expand on row 1 if it is not one of the two identical rows (you cam assume this, but say so). Explain briey the calculation that leads to 0. 1...
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