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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Matrices and Determinants (Set 2)

Take the ECAT Mathematics — Matrices and Determinants (Set 2): 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. For any square matrix A, the matrix A + Aᵗ is always:

    • A. skew-symmetric
    • B. symmetric ✓
    • C. diagonal
    • D. singular

    Answer: B. symmetric

    Explanation: (A + Aᵗ)ᵗ = Aᵗ + A = A + Aᵗ, so A + Aᵗ is symmetric. (A − Aᵗ is skew-symmetric.)

  2. Q2. The rank of the matrix [1 2 3; 2 4 6; 1 1 1] is:

    • A. 1
    • B. 3
    • C. 2 ✓
    • D. 0

    Answer: C. 2

    Explanation: R₂ = 2R₁, so R₂ → R₂ − 2R₁ gives a zero row; R₁ and R₃ are not proportional, so exactly 2 independent rows remain.

  3. Q3. For what value of k does the system x + 2y = 3, 2x + ky = 5 have no solution?

    • A. 6
    • B. 2
    • C. −4
    • D. 4 ✓

    Answer: D. 4

    Explanation: The coefficient determinant k − 4 must be 0, so k = 4. Then 2x + 4y = 5 contradicts 2x + 4y = 6 (twice the first equation), so the system is inconsistent.

  4. Q4. If A = [0 1; 1 0], then A² equals:

    • A. I (the 2 × 2 identity) ✓
    • B. 2A
    • C. O (the zero matrix)
    • D. A

    Answer: A. I (the 2 × 2 identity)

    Explanation: A² = [0·0 + 1·1, 0·1 + 1·0; 1·0 + 0·1, 1·1 + 0·0] = [1 0; 0 1] = I.

  5. Q5. For A = [1 2 3; 4 5 6; 7 8 9], the minor M₂₁ of the element in row 2, column 1 is:

    • A. 6
    • B. 3
    • C. −3
    • D. −6 ✓

    Answer: D. −6

    Explanation: Delete row 2 and column 1: M₂₁ = |2 3; 8 9| = 18 − 24 = −6.

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 and B are 3 × 3 matrices with |A| = 3 and |B| = −2, then |AB⁻¹| equals:

    • A. −2/3
    • B. −3/2
    • C. 3/2
    • D. −6

  2. Q7. If A and B are square matrices with AB = O (the zero matrix), then:

    • A. A = B
    • B. B must be the zero matrix
    • C. A or B need not be the zero matrix
    • D. A must be the zero matrix

  3. Q8. Using Cramer's rule, the value of y satisfying 3x + 2y = 12 and x − y = −1 is:

    • A. 3
    • B. 5
    • C. 2
    • D. −3

  4. Q9. The adjoint of the matrix [2 3; 1 4] is:

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

  5. Q10. If A = [1 2; 0 1] and B = [1 0; 3 1], then AB is:

    • A. [1 0; 3 1]
    • B. [7 3; 2 1]
    • C. [1 2; 3 7]
    • D. [7 2; 3 1]

  6. Q11. If 2X + [1 2; 3 4] = [3 6; 5 8], then X is:

    • A. [1 2; 1 2]
    • B. [2 4; 2 4]
    • C. [2 4; 4 6]
    • D. [1 2; 4 6]

  7. Q12. If every element of one row of a determinant is multiplied by k, the value of the determinant is:

    • A. multiplied by k
    • B. multiplied by kⁿ for an n × n determinant
    • C. unchanged
    • D. multiplied by k²

  8. Q13. The conjugate transpose (Ā)ᵗ of A = [1+i 2; 3i 4] is:

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

  9. Q14. If two rows of a square matrix are identical, its determinant is:

    • A. 0
    • B. 1
    • C. equal to the product of the diagonal elements
    • D. doubled

  10. Q15. The inverse of the matrix [3 2; 4 3] is:

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

  11. Q16. A non-homogeneous system AX = B of n linear equations in n unknowns with |A| ≠ 0 has:

    • A. infinitely many solutions
    • B. a unique solution
    • C. no solution
    • D. only the trivial solution X = O

  12. Q17. If A = [1 2; 3 4], then A² − 5A equals:

    • A. 5I
    • B. O
    • C. −2I
    • D. 2I

  13. Q18. If A is a 3 × 2 matrix, then the order of its transpose Aᵗ is:

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

  14. Q19. The value of the determinant |3 −2; 4 5| (rows separated by ';') is:

    • A. 7
    • B. 15
    • C. 23
    • D. −23

  15. Q20. The value of the determinant |1 a b+c; 1 b c+a; 1 c a+b| is:

    • A. abc
    • B. a + b + c
    • C. (a − b)(b − c)(c − a)
    • D. 0

  16. Q21. Which statement about elementary row operations on a matrix is correct?

    • A. They make every matrix symmetric
    • B. They do not change the rank of the matrix
    • C. They always change the determinant
    • D. They change the order of the matrix

  17. Q22. How many different orders (m × n) are possible for a matrix having exactly 12 elements?

    • A. 4
    • B. 6
    • C. 12
    • D. 3

  18. Q23. If A is a 3 × 3 matrix with |A| = 5, then A·(adj A) equals:

    • A. I
    • B. 25I
    • C. 125I
    • D. 5I

  19. Q24. If A is a 2 × 2 matrix with |A| = 2, then |3A| equals:

    • A. 18
    • B. 54
    • C. 6
    • D. 12

  20. Q25. A complex square matrix whose conjugate transpose equals the negative of the matrix itself is called:

    • A. Hermitian
    • B. orthogonal
    • C. skew-Hermitian
    • D. skew-symmetric

  21. Q26. For what value of x is the matrix [1 x; 3 2] symmetric?

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

  22. Q27. The value of the determinant |2 1 0; 1 3 1; 0 1 2| is:

    • A. 6
    • B. 10
    • C. 12
    • D. 8

  23. Q28. For non-singular square matrices A and B of the same order, (AB)⁻¹ equals:

    • A. A⁻¹ + B⁻¹
    • B. B⁻¹A⁻¹
    • C. BA
    • D. A⁻¹B⁻¹

  24. Q29. If A is a 3 × 3 matrix with |A| = −3, then |AAᵗ| equals:

    • A. −27
    • B. 9
    • C. −9
    • D. 27

  25. Q30. The matrix [x 2; 8 x] is singular when x equals:

    • A. ±4
    • B. 4 only
    • C. ±2
    • D. ±16