Determinants MCQS

SAMPLE M C Q II P U EXAM 2023 DETERMINANTS-4
DETERMINANTS
1. If A = kB, where A and B are square matrices of
order n, then | A| =
(A) 𝑘|B| (B)k n |B|
(C)𝑘 𝑛+1 |B| (D)𝑛𝑘|B|.
ANSWER:
0 sin 𝛼 − cos 𝛼
2. The value of determinant |− sin 𝛼 0 sin 𝛽 |
cos 𝛼 − sin 𝛽 0
(A) 0 (B)1
(C) sin 𝛼 (D) cos 𝛼.
ANSWER:
3 𝑥 3 2
3. If | |=| | then x is equal to
𝑥 1 4 1
(A) 2 (B) 4
(C) 8 (D)±2√2.
ANSWER:
2𝑥 4 2 4
4. If | | =| |, then value of x is
6 𝑥 5 1
(A) √3 (B)±√3
(C)±√6 (D) √6.

ANSWER:
5. If A is square matrix of order 3×3, then |𝑘A|is
equal to
(A) 𝑘|A| (B)k 2 |A|
(C)𝑘 3 |A| (D)3𝑘|A|
ANSWER:
1 2
6. If A= [ ], then |2A| is equal to
4 2
(A) 2|A| (B)3|A|
(C)4|A| (D)|A|.
ANSWER:
1 SATISH NAIK
1 0 1
5. If A=[0 1 2], then |3A| is equal to
0 0 4
(A) 27 (B) 4
(C) 54 (D)108.
ANSWER:
𝑥 2 6 2
18 𝑥 18 6
(A) 3 (B)±3
(C)±6 (D) 6.
ANSWER:
𝑥 3 2 3
2𝑥 5 4 5
(A) 2 (B)±2
(C-2 (D) 3.
ANSWER:
8. Which of the following statements is not true
(A) If any two rows or columns of a determinant are
identical, then the value of the determinant is zero.
(B) The value of the determinant remains no
changed if its rows and columns are interchanged.
(C) If any two rows (or columns) of a determinant
are interchanged, then the value of the determinant
changes in sign.
(D) If corresponding elements of any two rows (or
columns) of a determinant are proportional, then its
value is unchanged.
ANSWER:
9. Which of the following is correct
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a
matrix.
(C) Determinant is a number associated to a
square matrix.
(D) None of these.
ANSWER:
2 SATISH NAIK
2 3
10. Adjoint of a matrix A =[ ]
1 4
2 3 2 −3
(A) [ ] (B) [ ]
1 4 −1 4
4 3 4 −3
(C) [ ] (D) [ ].
1 2 −1 2
ANSWER:
11. If A be a non singular matrix of order 3, then
|adj A| is equal to
(A) |A| (B)|A|2
(C)|A|3 (D)3|A|.
ANSWER:
12. If A is an invertible matrix of order 2, then det
(𝐴−1 ) is equal to
1
(A) det (A) (B)det (A)
(C)0 (D)1.
ANSWER:
13. If A is a square matrix of order 2 and |A| = 3,

then | 𝐴−1 | =
2
(A) 3 (B)
3
1
(C (D) 12.
3
ANSWER:
14. If A is a square matrix of order n, then |adj(A) | =
(A) |A| (B)|A|n
(C)|A|𝑛−1 (D) 𝑛|A|.
ANSWER:
15. If A and B are invertible matrices, then which of
the following is not correct?
(A) A (adj A) = (adj A) A = A I
(B) A (adj A) = (adj A) A = |A|I
(C)(AB)−1=B−1 A−1
(D) |A| ≠ 0 and |B| ≠ 0 .
ANSWER:
3 SATISH NAIK
16. For a square matrix A in matrix equation

AX = B, Which of the following is not correct
(A) |A| ≠0, there exists unique solution
(B) |A| = 0 and (adj A) B ≠ 0, then there exists no
solution
(C) |A| = 0 and (adj A) B = 0, then system may or
may not be consistent
(D) |A| = 0 and (adj A) B = 0, then system may be
inconsistent.
ANSWER:
17 .If A is a matrix of order 3, such that
A(adjA) = 10 I then |adj A| =
(A) 10 (B)10 I
(C) 1 (D) 100.
ANSWER:
1 2 −1
18. If A = [1 𝑥 − 2 1 ] 𝑖𝑠 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑖𝑠
𝑥 1 1
(A) 2 (B) 3
(C) 1 (D) 0.
ANSWER:
19. If A is a square matrix of a order 3 and
|adjA|= 25, then |𝐴| is
1 1
(A) (B) 25 (C) 5 (D) .
25 5
ANSWER:
2 𝜆 −3
20. If A =[0 2 5 ] , then A−1 exists if
1 1 3
(A) 𝜆 = 2 (B)𝜆≠2
(C) 𝜆≠ -2 (D) 𝜆≠2 and 𝜆≠ -2.
ANSWER:
4 SATISH NAIK
𝐴𝑥 𝑥2 1 𝐴 𝐵 𝐶
21. Let ∆ = |𝐵𝑦 𝑦2 1| and ∆1= | 𝑥 𝑦 𝑧 | then
𝐶𝑧 𝑧2 1 𝑧𝑦 𝑧𝑥 𝑥𝑦
(A) ∆1= - ∆ (B) ∆1= ∆
(C) ∆1 ≠ ∆ (D) ∆1= 2∆.
ANSWER:
2 3
22. The inverse of the matrix [ ] is
4 6
2 3 2 −3
(A) [ ] (B)[ ]
4 6 −4 6
6 −3
(C) [ ] (D) Does not exists.
−4 2
ANSWER:
2 2
23. The inverse of the matrix [ ] is
4 3
1 3 −2 1 −3 2
(A) 2 [ ] (B) 2 [ ]
−4 2 4 −2
1 3 2 1 3 2
(C) 2 [ ](D) − 2 [ ].
4 2 4 2
ANSWER:
24. Consider the system of linear equations:
3𝑥 − 2𝑦 + 3𝑧=8, 2𝑥 + 𝑦 − 𝑧=1 and 4𝑥 − 3𝑦 + 2𝑧=4.The
system has
(A) exactly 3 solutions (B)a unique solution
(C) no solution (D) infinite number of solutions.
ANSWER:
25. If 𝐴2 − 4𝐴 + 𝐼 = 𝑂, then the inverse of A is

(A) A + I (B) A- 4I
(C) A - I (D) I – A.
ANSWER:
5 SATISH NAIK

Determinants MCQS

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SAMPLE M C Q II P U EXAM 2023 DETERMINANTS-4

(C)±√6 (D) √6.

13. If A is a square matrix of order 2 and |A| = 3,

16. For a square matrix A in matrix equation

25. If 𝐴2 − 4𝐴 + 𝐼 = 𝑂, then the inverse of A is

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