LINEAR ALGEBRA (MTS4 B04) CORE COURSE - BSc MATHEMATICS
IV SEMESTER
MULTIPLE CHOICE QUESTIONS
1. If A and B are square matrices of the same order, then
tr(AB) =
A. tr(A + B)
B. tr(A)tr(B)
C. tr(BA)
D. tr(A) + tr(B)
2. If A and B are square matrices of the same order, then
(AB)T =
A. AT B T
B. B T AT
C. AT + B T
D. (BA)T
3. If A and B are symmetric matrices of same order, then
A. AB is always symmetric
B. AB is never symmetric
C. AB is skew-symmetric
D. AB is symmetric if and only if AB = BA
4. For all sqaure matrices A and B, is it true that det(A+B) =
det(A) + det(B)
A. True
B. False
C. Cannot be determined
5. A matrix that is both symmetric and upper triangular must
be a
A. diagonal matrix
B. non-diagonal but symmetric
C. both A and B
D. none of the above
, 2
6. If A and B are invertible matrices with the same size, then
AB is invertible and (AB)−1 =
A. A−1 B −1
B. B −1 A−1
C. both A and B
D. none of the above
7. A matrix E is called . . . if it can be obtained from an iden-
tity matrix by performing a single elementary row opera-
tion.
A. equivalent matrix
B. echelon matrix
C. elementary matrix
D. row reduced matrix
8. Let A be an n × n matrix, and A is invertible. Then which
of the following statement is equivalent:
A. Ax = 0 has only the trivial solution.
B. A cannot be expressed as a product of elementary ma-
trices.
C. Ax = b is inconsistent for every n × 1 matrix b.
D. Ax = b has more than one solution for every n × 1
matrix b
9. A homogeneous linear system in n unknowns whose cor-
responding augmented matrix has a reduced row echelon
form with r leading 1’s has
A. n free variables
B. n - r free variables
C. r free variables
D. cannot be determined
10. A linear system is called consistent if it has
A. infinitely many solutions
B. no solution
, 3
C. at least one solution
D. none of the above
11. A consistent linear system of two equations in two un-
knowns has
A. exactly one solution
B. infinitely many solutions
C. exactly two solutions
D. either A or B
12. Which of the following matrices is in reduced row echelon
form.
1 0 0
A. 0 1 0
0 0 1
1 1 0
B. 0 1 0
0 0 0
C. both A and B
D. none
13. If TA : Rn → Rm and TB : Rn → Rm are matrix transforma-
tions, and if TA (x) = TB (x) for every vector x in Rn , then
A. A and B are equivalent but not equal
B. A and B are equal
C. A and B cannot be equal
D. cannot be determined
14. If S = {v1 , v2 , . . . , vn } is a set of vectors in a finite-
dimensional vector space V , then S is called a basis for V
if:
A. S spans V
B. S is linearly independent
C. either A or B
D. both A and B
IV SEMESTER
MULTIPLE CHOICE QUESTIONS
1. If A and B are square matrices of the same order, then
tr(AB) =
A. tr(A + B)
B. tr(A)tr(B)
C. tr(BA)
D. tr(A) + tr(B)
2. If A and B are square matrices of the same order, then
(AB)T =
A. AT B T
B. B T AT
C. AT + B T
D. (BA)T
3. If A and B are symmetric matrices of same order, then
A. AB is always symmetric
B. AB is never symmetric
C. AB is skew-symmetric
D. AB is symmetric if and only if AB = BA
4. For all sqaure matrices A and B, is it true that det(A+B) =
det(A) + det(B)
A. True
B. False
C. Cannot be determined
5. A matrix that is both symmetric and upper triangular must
be a
A. diagonal matrix
B. non-diagonal but symmetric
C. both A and B
D. none of the above
, 2
6. If A and B are invertible matrices with the same size, then
AB is invertible and (AB)−1 =
A. A−1 B −1
B. B −1 A−1
C. both A and B
D. none of the above
7. A matrix E is called . . . if it can be obtained from an iden-
tity matrix by performing a single elementary row opera-
tion.
A. equivalent matrix
B. echelon matrix
C. elementary matrix
D. row reduced matrix
8. Let A be an n × n matrix, and A is invertible. Then which
of the following statement is equivalent:
A. Ax = 0 has only the trivial solution.
B. A cannot be expressed as a product of elementary ma-
trices.
C. Ax = b is inconsistent for every n × 1 matrix b.
D. Ax = b has more than one solution for every n × 1
matrix b
9. A homogeneous linear system in n unknowns whose cor-
responding augmented matrix has a reduced row echelon
form with r leading 1’s has
A. n free variables
B. n - r free variables
C. r free variables
D. cannot be determined
10. A linear system is called consistent if it has
A. infinitely many solutions
B. no solution
, 3
C. at least one solution
D. none of the above
11. A consistent linear system of two equations in two un-
knowns has
A. exactly one solution
B. infinitely many solutions
C. exactly two solutions
D. either A or B
12. Which of the following matrices is in reduced row echelon
form.
1 0 0
A. 0 1 0
0 0 1
1 1 0
B. 0 1 0
0 0 0
C. both A and B
D. none
13. If TA : Rn → Rm and TB : Rn → Rm are matrix transforma-
tions, and if TA (x) = TB (x) for every vector x in Rn , then
A. A and B are equivalent but not equal
B. A and B are equal
C. A and B cannot be equal
D. cannot be determined
14. If S = {v1 , v2 , . . . , vn } is a set of vectors in a finite-
dimensional vector space V , then S is called a basis for V
if:
A. S spans V
B. S is linearly independent
C. either A or B
D. both A and B
