MATH 54 MIDTERM 1 (001)
PROFESSOR PAULIN
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Name and section:
GSI’s name:
Math 54 Midterm 1 (001) 9.10am - 10am
This exam consists of 5 questions. Answer the questions in thespaces provided.
1. (25 points) (a) Calculate the general solution to the linear system with the followingaugmented matrix:
⎛
⎜⎜⎝
1 0 −1 0 1 10 0 1 2 1 −10 0 0 0 1 30 0 0 0 0 0
⎞
⎟⎟⎠
Solution:
(b) Will the above coefficient matrix always give a consistent linear system? Justifyyour answer.
Solution:
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Math 54 Midterm 1 (001), Page 2 of 5 9.10am - 10am
2. (25 points) Calculate the determinant and inverse matrix of
⎛
⎜⎜⎝
0 0 −2 10 1 0 00 0 −1 11 1 −1 0
⎞
⎟⎟⎠.
Solution:
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Math 54 Midterm 1 (001), Page 3 of 5 9.10am - 10am
3. (25 points) Find all possible value of a, b, c such that
⎛
⎝11−1
⎞
⎠ is a solution to homogeneous
linear system
⎛
⎝a b c− 1 0b c c+ 2 02c −b a 0
⎞
⎠
Solution:
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Math 54 Midterm 1 (001), Page 4 of 5 9.10am - 10am
4. (25 points) (a) Let A be a 4× 5 matrix with the following properties:
The second column is non-zero and is a scalar multiple of the first. The thirdcolumn is not a scalar multiple of the first.
Write the echelon form matrices which are potentially row equivalent to A.
Solution:
(b) Let two matrices A and B satisfy the above conditions. If TA and TB are both ontomust A and B be row equivalent? Justify your answer.
Solution:
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Math 54 Midterm 1 (001), Page 5 of 5 9.10am - 10am
5. (25 points) Let T : R3 → R4 be a linear transformation such that
T
⎛
⎝101
⎞
⎠ =
⎛
⎜⎜⎝
01
t+ 1t+ 2
⎞
⎟⎟⎠ , T
⎛
⎝022
⎞
⎠ =
⎛
⎜⎜⎝
02t+ 22t+ 24t+ 4
⎞
⎟⎟⎠ , T
⎛
⎝0−10
⎞
⎠ =
⎛
⎜⎜⎝
−1−t−1−t
⎞
⎟⎟⎠ .
Calculate the standard matrix of T . For what values of t is T one-to-one? For whatvalue of t is T onto? Justify your answer.
Solution:
END OF EXAM