JEE (Main) Mathematics - MATRICES AND DETERMINANTS Multiple Choice Questions (MCQs)

JEE (Main) Mathematics - Matrices and Algebra of Matrices Questions

1. If A and B are two matrices of the same order, then which of the following properties is true?

   • a) A + B = B + A
   • b) A · B = B · A
   • c) A - B = B - A
   • d) A^T = B^T

   Answer: a) A + B = B + A

2. If A = [[2, 3], [-1, 4]], what is AT (the transpose of A)?

   • a) [[2, -1], [3, 4]]
   • b) [[2, 3], [-1, 4]]
   • c) [[3, 2], [4, -1]]
   • d) [[3, -1], [2, 4]]

   Answer: a) [[2, -1], [3, 4]]

3. A matrix is said to be symmetric if:

   • a) A^T = A
   • b) A^T = -A
   • c) AB = BA
   • d) A + A^T = 0

   Answer: a) A^T = A

4. Which of the following is a scalar matrix?

   • a) [[1, 0], [0, 1]]
   • b) [[2, 0], [0, 2]]
   • c) [[1, 2], [2, 1]]
   • d) [[0, 1], [1, 0]]

   Answer: b) [[2, 0], [0, 2]]

5. What is the determinant of the matrix A = [[2, 3], [1, 4]]?

   • a) 5
   • b) 8
   • c) 7
   • d) 6

   Answer: c) 7

6. If A is a diagonal matrix, then which of the following is true?

   • a) All diagonal elements are zero.
   • b) Non-diagonal elements are zero.
   • c) All elements are zero.
   • d) All elements are non-zero.

   Answer: b) Non-diagonal elements are zero.

7. The matrix [[1, 0], [0, -1]] is:

   • a) Symmetric but not skew-symmetric.
   • b) Skew-symmetric but not symmetric.
   • c) Neither symmetric nor skew-symmetric.
   • d) Both symmetric and skew-symmetric.

   Answer: d) Both symmetric and skew-symmetric.

8. The trace of a matrix is defined as:

   • a) The product of diagonal elements.
   • b) The sum of diagonal elements.
   • c) The determinant of the matrix.
   • d) The inverse of the matrix.

   Answer: b) The sum of diagonal elements.

9. If A · B ≠ B · A, then the matrices A and B:

   • a) Are both symmetric.
   • b) Are skew-symmetric.
   • c) Do not commute.
   • d) Are diagonal matrices.

   Answer: c) Do not commute.

10. The inverse of a diagonal matrix is:

    • a) Always a scalar matrix.
    • b) Always another diagonal matrix.
    • c) Always a null matrix.
    • d) Does not exist.

    Answer: b) Always another diagonal matrix.

11. A square matrix A is called idempotent if:

    • a) A² = I
    • b) A² = 0
    • c) A² = A
    • d) A² = -A

    Answer: c) A² = A

12. If A = [[1, 2], [3, 4]], what is 2A?

    • a) [[1, 4], [6, 8]]
    • b) [[2, 4], [6, 8]]
    • c) [[2, 4], [9, 16]]
    • d) [[2, 3], [3, 8]]

    Answer: b) [[2, 4], [6, 8]]

13. A matrix A is orthogonal if:

    • a) A^T = -A
    • b) A · A^T = I
    • c) A² = I
    • d) A + A^T = 0

    Answer: b) A · A^T = I

14. What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

    • a) 2
    • b) 3
    • c) 1
    • d) 0

    Answer: a) 2

15. The determinant of a skew-symmetric matrix of odd order is:

    • a) Zero
    • b) Non-zero
    • c) Positive
    • d) Negative

    Answer: a) Zero

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