Quiz5-Solutions

Quiz5-Solutions -...

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Unformatted text preview: Name:.......................................................................................... Math 331 - Quiz 5 - Solutions Thursday, October 13 3 −6 . Construct a 2 × 2 matrix B which has two different nonzero columns −2 4 such that AB is the zero matrix. 1. Let A = Solution: Write B in terms of its columns B= b1 b2 Then AB = Ab1 Ab2 So if AB = 0, b1 and b2 must be solutions to Ax = 0. So we solve 3 −6 0 −2 4 0 I ÷3 −→ − 1 −2 0 −2 4 0 I I +2I −− −→ 1 −2 0 000 2 where x2 is free. From the discussion 1 above, AB = 0 if the columns of B are solutions. Therefore, for example, In parametric vector form the solutions are x = x2 B= 42 21 is one possible B . 2. Suppose that the third column of B is all zeros. What can you say about the third column of AB ? Justify your answer. Solution: The third column of AB is Ab3 where b3 is the third column of B . Since b3 = 0, Ab3 = A0 = 0 So the third column of AB is also all zeros. ...
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This document was uploaded on 04/02/2012.

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