ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Matrices and Determinants (Set 4): 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. Evaluate the determinant |1 2 3; 4 5 6; 7 8 10| (rows separated by semicolons).
Answer: D. −3
Explanation: Expand along the first row: 1(50 − 48) − 2(40 − 42) + 3(32 − 35) = 2 + 4 − 9 = −3. (Without the 10 the rows would be dependent and the determinant would be 0.)
Q2. If A = [2 1; 5 3], the sum of all four entries of A⁻¹ is:
Answer: B. −1
Explanation: |A| = 6 − 5 = 1, so A⁻¹ = [3 −1; −5 2]. The entries add to 3 − 1 − 5 + 2 = −1.
Q3. A is a 3 × 3 matrix with |A| = 4. The value of |adj(2A)| is:
Answer: D. 1024
Explanation: |2A| = 2³·4 = 32. For a 3 × 3 matrix, |adj B| = |B|², so |adj(2A)| = 32² = 1024.
Q4. If A = [1 3; 0 1], the entry in the first row, second column of A¹⁵ is:
Answer: D. 45
Explanation: Aⁿ = [1 3n; 0 1] (by induction, since [1 3; 0 1][1 3m; 0 1] = [1 3(m + 1); 0 1]). For n = 15 the entry is 45.
Q5. If A = [2 −4; 1 −2], then A² is:
Answer: C. the zero matrix
Explanation: A² = [4 − 4, −8 + 8; 2 − 2, −4 + 4] = [0 0; 0 0]. A is a non-zero nilpotent matrix.
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Q6. Find the rank of the matrix with rows (1, 2, 3), (2, 4, 6), (1, 0, 1).
Q7. The solution (x, y, z) of x + y + z = 6, x − y + 2z = 5, 2x + y − z = 1 gives the product xyz equal to:
Q8. The homogeneous system kx + y + z = 0, x + ky + z = 0, x + y + kz = 0 has a non-trivial solution only for certain values of k. The sum of these values is:
Q9. If a = 1, b = 2, c = 3, the determinant |a b c; b c a; c a b| equals:
Q10. If |a b; c d| = 5, then |3a 3b; c + 2a d + 2b| equals:
Q11. For any square matrix A of order n, the product A · adj(A) equals:
Q12. If A is a skew-symmetric matrix of odd order, then |A| equals:
Q13. The matrix [2 1+i; z 3] is Hermitian when z equals:
Q14. If A = [2 1; 1 1] and AX = [5; 3], then x₁ − x₂ for X = [x₁; x₂] is:
Q15. The matrix A = [2 −2 −4; −1 3 4; 1 −2 −3] satisfies:
Q16. The matrix [3 −2; 4 −3] is best described as:
Q17. For A = [2 0 1; 3 1 4; 5 2 −1], the cofactor C₂₃ is:
Q18. The determinant of a matrix is 7. If rows 1 and 2 are interchanged and row 3 is then multiplied by 5, the new determinant is:
Q19. A and B are 3 × 3 matrices with |A| = 2 and |B| = 5. The value of |A⁻¹B²Aᵀ| is:
Q20. If D = diag(2, 3), the sum of the diagonal entries of D⁴ is:
Q21. The determinant of [2 5 1 7; 0 3 9 2; 0 0 −1 4; 0 0 0 5] is:
Q22. The determinant |1 x x²; 1 2 4; 1 3 9| vanishes for how many values of x, and what is their sum?
Q23. For the system 2x + 3y = 8, x − y = −1, the determinant Dᵧ used in Cramer's rule is:
Q24. If [x + y 2; 3 x − y] = [5 2; 3 1], then xy equals:
Q25. A is a 2 × 2 matrix with |A| = 5. The value of |adj A| is:
Q26. Which of the following is true for all n × n matrices A and B?
Q27. The system x + 2y − z = 3, 2x + 4y − 2z = 7 has:
Q28. For A = [k 2; 8 k], A has no inverse for the positive value k equal to:
Q29. For A = [1 2 0; 0 1 3; 0 0 1], the entry in the first row and third column of A⁻¹ is:
Q30. If A = [1 2; 0 1] and I is the identity matrix, then A³ − 3A² + 3A − I equals: