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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.
 Fall '09
 Math

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