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Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the matrix product AB, if it is defined.
- A = 1 3^ -^3 3 0 5
, B =
A)
B)
C) AB is undefined. D)
Perform the matrix operation.
- Let A = (^) - 5 2 and B = 1 0. Find 2A + 3B. A) (^) - 10 4 B) (^) - 2 2 C) (^) - 9 4 D) (^) - 7 4
Find the inverse of the matrix, if it exists.
- A = - 5 4 0 4
A)
B)
C)
D)
Decide whether or not the matrices are inverses of each other.
4) -^2
and
A) Yes B) No
Determine whether the matrix is invertible.
- 6 1 3 4 A) Yes B) No
A) No B) Yes
Compute the determinant of the matrix by cofactor expansion.
A) 1084 B) - 286 C) 286 D) 146
A) 0 B) - 80 C) 80 D) - 40
A) - 9 B) - 36 C) 0 D) 36
A) - 200 B) 100 C) - 100 D) - 50
Calculate the area of the parallelogram with the given vertices.
- (0, 0), (2, 6), (11, 8), (9, 2) A) 52 B) 100 C) 50 D) 49
Determine the values of the parameter s for which the system has a unique solution, and describe the solution.
- sx1 - 5sx2 = 3 3x1 - 15sx2 = 5
A) s ≠ 1; x1 = 4 15(s - 1)(s + 1)
and x2 = 9 -^ 5s 15(s - 1)(s + 1)
B) s ≠ 1; x1 = 14 3(s + 1)
and x2 = 9 +^ 5s 15s(s + 1)
C) s ≠ 0, 1; x1 = 4 3(s - 1)
and x2 = 9 -^ 5s 15s(s - 1)
D) s ≠ ± 1; x1 = 4 3(s + 1)
and x2 = 9 -^ 5s 15s(s + 1)
Find a basis for the column space of the matrix.
- Find a basis for Col B where
B =
A)
B)
C)
D)
Determine which of the sets of vectors is linearly independent.
- A: The set p 1, p 2, p 3 where p 1(t) = 1, p 2(t) = t2, p 3(t) = 3 + 3t
B: The set p 1, p 2, p 3 where p 1(t) = t, p 2(t) = t2, p 3(t) = 2t + 3t
C: The set p 1, p 2, p 3 where p 1(t) = 1, p 2(t) = t2, p 3(t) = 3 + 3t + t
A) C only B) all of them C) A only D) B only E) A and C
Find the vector x determined by the given coordinate vector [x] B and the given basis B****.
- B = -^1
, [x] B = -^2 3 A) 2 3
B)
C)
D)
Find the coordinate vector [x] B of the vector x relative to the given basis B****.
- b 1 = 3 2
, b 2 = 3
, x = -^9
, and B = b 1, b 2
A)
B)
C)
D)
Solve the problem.
- Let H =
a + 2b + 2d c + d
: a, b, c, d in ℛ
Find the dimension of the subspace H. A) dim H = 3 B) dim H = 4 C) dim H = 2 D) dim H = 1
Find the dimensions of the null space and the column space of the given matrix.
- A = 1 -^3 -^5 3
- 2 1 3 - 4 1 A) dim Nul A = 4, dim Col A = 1 B) dim Nul A = 2, dim Col A = 3 C) dim Nul A = 3, dim Col A = 2 D) dim Nul A = 3, dim Col A = 3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
- Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all possible constants on the right side of the equation. Is it possible to find 4 nonzero solutions of the associated homogeneous system that are linearly independent? Explain.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
- If A is a 7 × 9 matrix, what is the smallest possible dimension of Nul A? A) 9 B) 7 C) 2 D) 0
