ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Matrices and Determinants (Set 3): 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 value of the determinant |1 2 3; 0 4 5; 1 0 6| (rows separated by ';') is:
Answer: C. 22
Explanation: Expanding along the first column: 1(24 − 0) − 0 + 1(10 − 12) = 24 − 2 = 22.
Q2. If A = [1 1; 2 3], then A² − 4A equals:
Answer: D. −I
Explanation: A² = [3 4; 8 11] and 4A = [4 4; 8 12]; the difference is [−1 0; 0 −1] = −I (Cayley–Hamilton: A² − 4A + I = O).
Q3. If A is a 3 × 3 matrix with |A| = 7, then |adj(2A)| equals:
Answer: A. 3136
Explanation: |2A| = 2³|A| = 56. For a 3 × 3 matrix |adj B| = |B|², so |adj(2A)| = 56² = 3136.
Q4. If 2X + [1 2; 3 4] = [5 0; 1 6], then X is:
Answer: B. [2 −1; −1 1]
Explanation: 2X = [4 −2; −2 2], so X = [2 −1; −1 1].
Q5. If A = [0 1; −1 0], then A²⁰²⁶ equals:
Answer: D. −I
Explanation: A² = −I and A⁴ = I. Since 2026 = 4 × 506 + 2, A²⁰²⁶ = A² = −I.
Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.
Q6. The matrix A = [1 −1; 1 −1] satisfies:
Q7. The matrix [a 3; 2 a+1] is singular for two values of a. Their sum is:
Q8. The rank of the matrix [1 2 3; 2 4 7; 3 6 10] is:
Q9. The solution of x + y + z = 6, x − y + z = 2, 2x + y − z = 1 has z equal to:
Q10. The system x + 2y = 3, 2x + 4y = k has infinitely many solutions when k equals:
Q11. Using Cramer's rule for 2x + 3y = 8, x − y = −1, the value of x is:
Q12. If A = [1 2; 3 4] and B = [0 1; 1 0], the trace of AB is:
Q13. The symmetric part ½(A + Aᵀ) of A = [1 4; 2 3] has (1,2) entry equal to:
Q14. The determinant of any 3 × 3 skew-symmetric matrix is:
Q15. The matrix [2 3+4i; x 5] is Hermitian when x equals:
Q16. The matrix [3/5 4/5; x 3/5] is orthogonal (rows orthogonal unit vectors) for x equal to:
Q17. If A = [2 0; 0 3], the sum of all entries of A⁵ is:
Q18. The determinant |1 1 1; 1 2 3; 1 4 9| equals:
Q19. A 3 × 3 matrix A has |A| = 3. B is formed by swapping R₁ and R₂, then multiplying R₃ by 2, then adding R₁ to R₂. Then |B| equals:
Q20. The determinant |x x+1 x+2; x+3 x+4 x+5; x+6 x+7 x+8| equals:
Q21. For A = [1 2 3; 4 5 6; 7 8 10], the cofactor C₂₁ is:
Q22. If A = [1 2; 0 1], then the (1,2) entry of A⁵⁰ is:
Q23. If [2 1; 5 3][x; y] = [3; 8], then x + 2y equals:
Q24. The homogeneous system x + 2y + z = 0, 2x + y + 3z = 0, x + y + kz = 0 has a non-trivial solution when k equals:
Q25. The determinant of the upper triangular matrix [2 5 1; 0 3 7; 0 0 4] is:
Q26. If A is a square matrix with A² = A, then (I + A)² equals:
Q27. For square matrices A and B of the same order, which expansion is always correct?
Q28. If A is a 3 × 3 matrix with |A| = 7, then |−2A| equals:
Q29. In a 3 × 4 matrix with aᵢⱼ = i + j, the sum of all entries is:
Q30. If A = [3 1; 5 2], the sum of the entries of A⁻¹ is: