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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Matrices and Determinants (Set 1)

Take the ECAT Mathematics — Matrices and Determinants (Set 1): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. If A is a 2 × 3 matrix and B is a 3 × 4 matrix, the order of AB is:

    • A. 2 × 4 ✓
    • B. 2 × 3
    • C. 3 × 3
    • D. 4 × 2

    Answer: A. 2 × 4

    Explanation: For AB, the inner orders (3 and 3) must match; the product has the outer orders 2 × 4.

  2. Q2. The value of the determinant |2 3; 1 4| (rows separated by ';') is:

    • A. −5
    • B. 11
    • C. 5 ✓
    • D. 8

    Answer: C. 5

    Explanation: |2 3; 1 4| = 2·4 − 3·1 = 8 − 3 = 5.

  3. Q3. The inverse of the matrix [2 1; 5 3] is:

    • A. [3 1; 5 2]
    • B. [2 −5; −1 3]
    • C. [−3 1; 5 −2]
    • D. [3 −1; −5 2] ✓

    Answer: D. [3 −1; −5 2]

    Explanation: |A| = 6 − 5 = 1, so A⁻¹ = adj A = [3 −1; −5 2].

  4. Q4. For what value of k is the matrix [k 4; 3 6] singular?

    • A. 4.5
    • B. 2 ✓
    • C. 8
    • D. −2

    Answer: B. 2

    Explanation: Singular means determinant zero: 6k − 12 = 0 ⇒ k = 2.

  5. Q5. The determinant of the matrix [1 2 3; 0 4 5; 0 0 6] is:

    • A. 24 ✓
    • B. 11
    • C. 0
    • D. 6

    Answer: A. 24

    Explanation: The matrix is upper triangular, so its determinant is the product of the diagonal entries: 1 × 4 × 6 = 24.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

  1. Q6. If A is a 3 × 3 matrix with |A| = 5, then |2A| equals:

    • A. 10
    • B. 20
    • C. 40
    • D. 30

  2. Q7. For any square matrix A, the determinant of its transpose satisfies:

    • A. |Aᵗ| = 1/|A|
    • B. |Aᵗ| = |A|²
    • C. |Aᵗ| = −|A|
    • D. |Aᵗ| = |A|

  3. Q8. If two rows of a determinant are interchanged, the value of the determinant:

    • A. becomes zero
    • B. remains unchanged
    • C. is doubled
    • D. changes sign only

  4. Q9. If A = [1 2; 3 4], then A² is:

    • A. [2 4; 6 8]
    • B. [7 15; 10 22]
    • C. [1 4; 9 16]
    • D. [7 10; 15 22]

  5. Q10. For matrices A and B conformable for multiplication, (AB)ᵗ equals:

    • A. BA
    • B. (BA)ᵗ
    • C. BᵗAᵗ
    • D. AᵗBᵗ

  6. Q11. Which of the following matrices is skew-symmetric?

    • A. [0 2 −3; 2 0 −4; −3 4 0]
    • B. [0 2 3; 2 0 4; 3 4 0]
    • C. [1 2 −3; −2 1 4; 3 −4 1]
    • D. [0 2 −3; −2 0 4; 3 −4 0]

  7. Q12. The diagonal elements of every skew-symmetric matrix are:

    • A. all zero
    • B. all equal but non-zero
    • C. all equal to 1
    • D. purely imaginary

  8. Q13. The adjoint of the matrix [a b; c d] is:

    • A. [a −c; −b d]
    • B. [d b; c a]
    • C. [−d b; c −a]
    • D. [d −b; −c a]

  9. Q14. If A is a 3 × 3 matrix with |A| = 4, then |adj A| equals:

    • A. 12
    • B. 64
    • C. 4
    • D. 16

  10. Q15. Using Cramer's rule, the value of x satisfying 2x + y = 5 and x − y = 1 is:

    • A. 3
    • B. 1
    • C. −2
    • D. 2

  11. Q16. A homogeneous system AX = O of n equations in n unknowns has a non-trivial solution if and only if:

    • A. |A| ≠ 0
    • B. A is symmetric
    • C. |A| = 0
    • D. A is the identity matrix

  12. Q17. The rank of the matrix [1 2; 2 4] is:

    • A. 4
    • B. 2
    • C. 0
    • D. 1

  13. Q18. For A = [1 2 3; 4 5 6; 7 8 10], the cofactor A₁₂ of the element in row 1, column 2 is:

    • A. −8
    • B. 8
    • C. 2
    • D. −2

  14. Q19. A square matrix A with complex entries is called Hermitian if:

    • A. Ā = A
    • B. (Ā)ᵗ = −A
    • C. (Ā)ᵗ = A
    • D. Aᵗ = −A

  15. Q20. If ω is a complex cube root of unity, the value of |1 ω ω²; ω ω² 1; ω² 1 ω| is:

    • A. −3ω
    • B. 0
    • C. 3
    • D. 1

  16. Q21. The inverse of the diagonal matrix [2 0; 0 3] is:

    • A. [3 0; 0 2]
    • B. [−2 0; 0 −3]
    • C. [1/3 0; 0 1/2]
    • D. [1/2 0; 0 1/3]

  17. Q22. The value of x for which |x 2; 3 x−1| = 0 is:

    • A. −3 or 2
    • B. 3 or −2
    • C. 6 or −1
    • D. 2 or 3

  18. Q23. The determinant |1 1 1; a b c; a² b² c²| is equal to:

    • A. abc
    • B. (a − b)(b − c)(a − c)
    • C. (a + b)(b + c)(c + a)
    • D. (a − b)(b − c)(c − a)

  19. Q24. The trace of the matrix [3 1 2; 0 −1 5; 4 2 7] is:

    • A. 23
    • B. 9
    • C. 11
    • D. 10

  20. Q25. Which of the following is a scalar matrix?

    • A. [5 5; 5 5]
    • B. [5 0; 0 5]
    • C. [5 0; 0 3]
    • D. [0 5; 5 0]

  21. Q26. If A is a non-singular 2 × 2 matrix with |A| = 3, then |A⁻¹| is:

    • A. 1/3
    • B. 3
    • C. 1/9
    • D. −3

  22. Q27. Which statement about square matrices A and B of the same order is correct in general?

    • A. AB need not equal BA
    • B. (A + B)² = A² + 2AB + B² always
    • C. AB = O implies A = O or B = O
    • D. AB = BA always

  23. Q28. Applying the row operation R₂ → R₂ − 3R₁ to [1 2; 3 4] gives:

    • A. [1 2; 0 2]
    • B. [1 2; 0 −2]
    • C. [−2 −2; 3 4]
    • D. [1 2; 0 −10]

  24. Q29. If A = [1 0; 2 1] and B = [3 1; 0 2], then 2A − B is:

    • A. [−1 −1; 4 0]
    • B. [1 1; 4 0]
    • C. [−1 1; 4 0]
    • D. [−1 −1; 2 −1]

  25. Q30. The solution of the matrix equation AX = B, where A = [1 2; 3 5] and B = [1; 2] (column matrix), is X =

    • A. [−1; 1]
    • B. [−1; −1]
    • C. [1; 0]
    • D. [1; −1]