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Unformatted text preview: Solutions to quiz # 3 (September 23)
1. Bringing the matrix to the 1
a
3 reduced rowechelon form, we obtain 11
1
1
1
2 4 −→ 0 2 − a 4 − a .
3a
0
0
a−3 If a = 2, 3 we can proceed further 1
1
1
1 0 2 − a 4 − a −→ 0
0
0
a−3
0 1
1
0 1 and the rank of the matrix is 3.
If a = 2, we have the matrix 11 1
11 0 0 2 −→ 0 0
0 0 −1
00
and the rank of the matrix is 2.
If a = 3, we have the matrix 11 0 −1
00 0
100
0 −→ 0 1 0 1
001 11
4−a −→ 0 1
2−a
1
00 1
10 −→ 0 1
1
0
00 0
1
0 2
−1 0 and the rank of the matrix is 2.
Answer: The rank is 2 when a = 2 or when a = 3.
2. We observe that vector
vector 1
0 is mapped to vector 0
−1 −1
. Hence the matrix of the transformation is
0 y
0
1 y=x −1
0 and vector
0
−1 1
0
x 0
0
−1 Answer: The matrix of the transformation is
1 0
−1 −1
.
0 −1
.
0 0
1 is mapped to ...
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 Fall '08
 Conger
 Linear Algebra, Algebra

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