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