ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Matrices and Determinants (Set 5): 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 (2, −1, 0), (1, 3, 4), (0, 5, −2) is:
Answer: A. −54
Explanation: Expanding along the first row: 2(3·(−2) − 4·5) − (−1)(1·(−2) − 4·0) + 0 = 2(−26) + (−2) = −54.
Q2. If A = (1 1; 0 1) (rows separated by semicolons), the sum of all entries of A²⁵ is:
Answer: C. 27
Explanation: Aⁿ = (1 n; 0 1) by induction, so A²⁵ = (1 25; 0 1) and the entries sum to 1 + 25 + 0 + 1 = 27.
Q3. If D = (2 0; 0 3), the trace of D⁵ is:
Answer: A. 275
Explanation: D⁵ = (32 0; 0 243), whose trace is 32 + 243 = 275. (Note tr(D)⁵ = 3125, not the same.)
Q4. A is a 3 × 3 matrix with |A| = 4. Then |adj(2A)| equals:
Answer: D. 1024
Explanation: |2A| = 2³·4 = 32 and |adj B| = |B|ⁿ⁻¹ = |B|² for 3 × 3, so |adj(2A)| = 32² = 1024.
Q5. The determinant of (1 1 1; 1 2 4; 1 3 9), rows separated by semicolons, equals:
Answer: C. 2
Explanation: This is a Vandermonde determinant for 1, 2, 3: (2 − 1)(3 − 1)(3 − 2) = 2.
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Q6. For A = (1 2; 3 4), the matrix A² − 5A equals:
Q7. If A and B are 3 × 3 matrices with |A| = 2 and |B| = 3, then |A⁻¹BᵀA| equals:
Q8. The rank of the matrix (1 2 3; 2 4 6; 1 0 1), rows separated by semicolons, is:
Q9. The system x + y + z = 6, x + 2y + 3z = 14, x + 2y + λz = μ has infinitely many solutions when:
Q10. The system x + 2y + 3z = 0, 2x + ky + z = 0, 3x + y − z = 0 has a non-trivial solution when k equals:
Q11. If A = (cos θ −sin θ; sin θ cos θ) with θ = π/6, then A⁶ equals:
Q12. The matrix (3 x + iy; 2 − 5i 4) is Hermitian. Then x + y equals:
Q13. If A = (3/5 p; −4/5 q) is a real orthogonal matrix with |A| = 1, then p + q equals:
Q14. For the matrix A = (2 −2; 1 −1), the trace of A²⁰²⁶ is:
Q15. If N = (1 −1; 1 −1), then (I − N)⁻¹ equals:
Q16. A is a 3 × 3 matrix with A·adj(A) = 5I. Then |adj A| equals:
Q17. The cofactor of the entry in row 2, column 1 of (1 2 3; 0 4 5; 1 0 6), rows separated by semicolons, is:
Q18. The determinant of (1 a b+c; 1 b c+a; 1 c a+b), rows separated by semicolons, equals:
Q19. The sum of all values of x satisfying |x + 1 3; 5 x + 3| = 0 is:
Q20. A 3 × 3 matrix A has |A| = 5. B is obtained from A by interchanging R₁ and R₃, then multiplying R₂ by 4, then replacing R₁ by R₁ + R₃. Then |B| equals:
Q21. For A = (1 5; 3 7), the matrix (A − Aᵀ)² equals:
Q22. If A = (2 1; 1 2), then A² − 4A + 3I = 0, so A⁻¹ equals:
Q23. How many 2 × 2 matrices whose entries are all 0 or 1 are singular?
Q24. If A⁻¹ = (2 −1; −3 2) and AX = (1; 2), then x + 2y for X = (x; y) is:
Q25. Which matrices B = (a b; c d) satisfy AB = BA for A = (2 1; 0 2)?
Q26. If A = (1 1; 1 0), the entry in the first row and first column of A⁷ is:
Q27. By Cramer's rule, y for x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 is:
Q28. For A = (4 7; 1 2), the sum of all four entries of A⁻¹ is:
Q29. Write A = (2 5; 1 4) as the sum of a symmetric and a skew-symmetric matrix. The (1, 2) entry of the skew-symmetric part is:
Q30. A real skew-symmetric matrix of order 3 always has determinant: