Math 1104C, 1104J: Test # 1, Due Feb. 9 11:59, 2022
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• The assignment has 8 questions for a total of 40 points.
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1. (5 points) Use Gauss-Jordan elmination to find all solutions to the linear system:
x1 + 3x2 + 2x3 + 4x4 = 2
x3 + x4 = 2
2x1 + 6x2 + 3x3 + 7x4 = 2
2. (5 points) Let
A =
XXXXXXXXXX 0
XXXXXXXXXX
~b =
11
1
Find the reduced row echelon form of (A|~b). Use it to find a solutions to A~x = ~b.
3. (5 points) Solve the system A~x = ~0 using the matrix A from question 5.
4. (6 points) Let A =
(
1 0 −1
1 1 1
)
. Is ~b =
(
3
2
)
in the span of the columns of A? Is ~w =
(
−1 1 1
)
in
the span of the rows of A?
5. (6 points) Let
A =
(
3 −1 2
−1 2 4
)
, B =
1 2−1 −2
0 3
, C = (0 1
1 0
)
Evaluate the following expressions if they make sense.
(a) CB
(b) BC
(c) AB + C
(d) BA + C
(e) CT + C
(f) B + AT
6. (3 points) Give an example of a matrix M where MMT and MTM are not equal matrices.
7. (5 points) Is
P =
(
8 1
5 −2
)
a linear combination of the three matrices
Q1 =
(
1 1
1 1
)
, Q2 =
(
0 1
−1 0
)
, Q3 =
(
1 0
0 −1
)
8. (5 points) Find the inverse of the following matrix: 1 1 −10 1 1
0 1 −1
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