ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Matrices and Determinants (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. A 3 × 3 matrix A has entries aᵢⱼ = 2i + 3j. Its determinant is
Answer: C. 0
Explanation: Row i is (2i + 5, 2i + 8, 2i + 11), so R₃ − R₂ = R₂ − R₁ = (2, 2, 2). The rows are linearly dependent and |A| = 0 (the rank is 2).
Q2. The 3 × 3 matrix A has entries aᵢⱼ = min(i, j). Its determinant is
Answer: A. 1
Explanation: A = [[1, 1, 1], [1, 2, 2], [1, 2, 3]]. R₂ − R₁ and R₃ − R₂ give an upper triangular matrix with diagonal 1, 1, 1, so |A| = 1.
Q3. If A = [[2, −1], [3, 4]], the trace of A² is
Answer: A. 14
Explanation: A² = [[1, −6], [18, 13]], so the trace is 1 + 13 = 14. (Squaring the trace, 36, or the diagonal entries, 4 + 16 = 20, would be wrong.)
Q4. The system x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent for exactly two values of λ. Their sum is
Answer: A. 3
Explanation: The coefficient determinant is 0. Combining equations 1, 2, 3 as 2·E₁ − 3·E₂ + E₃ eliminates all variables: 0 = 2 − 3λ + λ², so λ² − 3λ + 2 = 0, giving λ = 1, 2 with sum 3.
Q5. The 4 × 4 matrix whose (i, j) entry is i + j has rank
Answer: D. 2
Explanation: Every row equals the first row plus a constant multiple of (1, 1, 1, 1) (R₃ = 2R₂ − R₁, R₄ = 3R₂ − 2R₁), so only R₁ and R₂ are independent. Rank = 2.
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Q6. If A = [[1, 2], [2, 1]], the entry in the first row and second column of A⁴ is
Q7. N = [[0, 1, 0], [0, 0, 1], [0, 0, 0]]. The sum of all nine entries of (I + N)⁻¹ is
Q8. The value of the determinant with rows (101, 102, 103), (104, 105, 106), (107, 108, 109) is
Q9. For real a, b, c, the determinant with rows (a − b, b − c, c − a), (b − c, c − a, a − b), (c − a, a − b, b − c) equals
Q10. A shop sells notebooks (x rupees), pens (y rupees) and rulers (z rupees). One notebook, one pen and one ruler cost 9 in all; two notebooks and a pen cost 9 more than a ruler; and a notebook plus 3 rulers cost 4 more than 2 pens. The product xyz is
Q11. For A = [[4, 1], [3, 2]] one has A² − 6A + 5I = O. Using this, 5A⁻¹ equals
Q12. For L = [[1, 0, 0], [2, 1, 0], [3, 4, 1]], the entry in the third row and first column of L⁻¹ is
Q13. How many independent entries are needed to specify a 4 × 4 real symmetric matrix completely?
Q14. How many 4 × 4 skew-symmetric matrices have every entry in {−1, 0, 1}?
Q15. A is a real 2 × 2 matrix with A² = I that is neither I nor −I. Then |A| equals
Q16. If A is a Hermitian matrix, then the matrix iA (with i = √−1) is
Q17. A 3 × 3 matrix A has integer entries, all even. The determinant of A must be divisible by
Q18. A 3 × 3 matrix A has |A| = 6. B is formed from A by replacing its rows R₁, R₂, R₃ with R₁ + R₂, R₂ + R₃, R₃ + R₁. Then |B| equals
Q19. For a 2 × 2 matrix A, adj A = [[4, −2], [−3, 1]]. Then A equals
Q20. A is a 2 × 3 matrix and B is a 3 × 2 matrix. Which statement is always true?
Q21. The homogeneous system x + 2y − z = 0, 2x + y + z = 0 has non-trivial solutions. If x = 1 in such a solution, then z equals
Q22. If X[[1, 2], [3, 5]] = [[3, 1], [2, 0]] (that is, XA = B with A = [[1, 2], [3, 5]]), then X equals
Q23. The points (2, 3), (5, k) and (8, −3) are collinear. Using the area determinant, k equals
Q24. The determinant with rows (1, 1, 1), (1, 2, 8), (1, 3, 27) (entries 1, a, a³ for a = 1, 2, 3) equals
Q25. If A is a real skew-symmetric matrix, then A² is
Q26. If A is a symmetric matrix and B is a skew-symmetric matrix of the same order, then AB − BA is
Q27. The sum of all real values of x for which |[[x, 2], [3, x + 1]]| = 4 is
Q28. For square matrices A and B of the same order, (A + B)² = A² + 2AB + B² holds if and only if
Q29. For a 3 × 3 matrix A = [aᵢⱼ] with cofactors Cᵢⱼ, the sum a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃ equals
Q30. The number of distinct real values of x for which |[[x, 1, 1], [1, x, 1], [1, 1, x]]| = 0 is