ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Matrices and Determinants (Set 6): 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. The determinant of the matrix with rows (1, 2, 3), (4, 5, 6), (7, 8, 10) is:
Answer: C. −3
Explanation: Expanding along the first row: 1(50 − 48) − 2(40 − 42) + 3(32 − 35) = 2 + 4 − 9 = −3.
Q2. If A is a 3 × 3 matrix with det A = 2, then det(adj(adj A)) equals:
Answer: B. 16
Explanation: For n = 3, adj(adj A) = (det A)ⁿ⁻² A = 2A, and det(2A) = 2³ · 2 = 16.
Q3. The rank of the 3 × 4 matrix with rows (1, 2, 0, 1), (2, 4, 1, 5), (3, 6, 1, 6) is:
Answer: A. 2
Explanation: Row 3 = Row 1 + Row 2, so the three rows span only two independent directions, and rows 1 and 2 are not proportional. Hence the rank is 2.
Q4. If A is a 2 × 2 matrix with A² = A, then (A + I)⁵ equals:
Answer: C. I + 31A
Explanation: Because A is idempotent, Aᵏ = A for k ≥ 1, so (A + I)⁵ = I + (C(5,1) + C(5,2) + C(5,3) + C(5,4) + C(5,5))A = I + (2⁵ − 1)A = I + 31A.
Q5. If A = [[1, 1], [−1, −1]] and I is the 2 × 2 identity, then (I − A)⁻¹ equals:
Answer: D. [[2, 1], [−1, 0]]
Explanation: A² = 0, so (I − A)(I + A) = I − A² = I and (I − A)⁻¹ = I + A = [[2, 1], [−1, 0]].
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Q6. The value of the determinant with rows (a, b, c), (b, c, a), (c, a, b) when a = 1, b = 2, c = 3 is:
Q7. The system x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ has infinitely many solutions for:
Q8. Using Cramer's rule or elimination, the solution (x, y, z) of x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 has z equal to:
Q9. For A = [[2, 1], [1, 1]], the trace of A⁻¹ is:
Q10. If [[1, 2], [3, 5]] X = [[3, 1], [2, 0]], then X equals:
Q11. If A and B are 3 × 3 matrices with det A = 3 and det B = −2, then det(2 A Bᵀ) equals:
Q12. If B is a non-singular 4 × 4 matrix, A is non-singular and det B = 5, then det(A⁻¹ B² A) equals:
Q13. If A = [[0, a, b], [−a, 0, c], [−b, −c, 0]] with real a, b, c, then det A equals:
Q14. For A = [[1, 3], [5, 7]], the skew-symmetric part ½(A − Aᵀ) equals:
Q15. The matrix [[i, a], [−1 + 2i, 3i]] is skew-Hermitian when a equals:
Q16. How many 2 × 2 matrices with every entry equal to 0 or 1 are non-singular?
Q17. How many symmetric 3 × 3 matrices can be formed with every entry chosen from {−1, 0, 1}?
Q18. If det A = 4 and B is obtained from A by interchanging two rows, then adding 5 times row 1 to row 3, then doubling row 2, det B equals:
Q19. The rows of M = [[1, 2, a], [2, 4, 6], [1, 2, 3]] are linearly dependent and rank M = 1 exactly when a equals:
Q20. The matrix A = [[1, 2], [3, 4]] satisfies A² − 5A − 2I = 0. The entry in row 2, column 1 of A⁻¹ is:
Q21. If A = [[cos θ, sin θ], [−sin θ, cos θ]] and θ = π/12, then A¹² equals:
Q22. If M is a 3 × 3 matrix with det M = −3, then det(adj(2M)) equals:
Q23. If A⁻¹ = [[1, 2], [0, 1]] and B⁻¹ = [[1, 0], [3, 1]], then the (2, 2) entry of (AB)⁻¹ is:
Q24. The homogeneous system kx + y + z = 0, x + ky + z = 0, x + y + kz = 0 has non-trivial solutions for exactly two values of k. Their sum is:
Q25. The determinant with rows (1, 1, 1), (1, 2, 4), (1, 4, 16) equals:
Q26. The area of the triangle with vertices (1, 2), (4, 6) and (7, −2) in square units is:
Q27. The matrix A = ⅓[[1, 2, 2], [2, 1, −2], [2, −2, 1]] satisfies AAᵀ = I. Its determinant is:
Q28. If A = [[1, 1], [0, 1]], then the (1, 2) entry of A + A² + A³ + … + A¹⁰ is:
Q29. For any two 2 × 2 matrices A and B, the trace of AB − BA equals:
Q30. The matrix with rows (x + 1, 2, 3), (1, x + 2, 3), (1, 2, x + 3) is singular for x = 0 and for exactly one non-zero value of x. That value is: