Engineering Mathematics Matrices MCQ

Solve inverse of matrix MCQ, transpose of matrix MCQ, trace of matrix MCQ, types of matrix MCQ types questions with their answers.

1. The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?

1. PQ = I but QP ≠ I
2. QP = I but PQ ≠ I
3. PQ = I and QP = I
4. PQ – QP = I

2. If $A=\begin{bmatrix} 1 & 5\\ 6 & 2 \end{bmatrix} \; and \; \begin{bmatrix} 3 & 7\\ 8 & 4 \end{bmatrix}$, AB^T is equal to

1. $\begin{bmatrix} 38 & 28\\ 32 & 56 \end{bmatrix}$
2. $\begin{bmatrix} 3 & 40\\ 42 & 8 \end{bmatrix}$
3. $\begin{bmatrix} 43 & 27\\ 34 & 50 \end{bmatrix}$
4. $\begin{bmatrix} 38 & 32\\ 28 & 56 \end{bmatrix}$

3. The matrix $\begin{bmatrix} a & 0 & 3 & 7\\ 2 & 5 & 1 & 3\\ 0 & 0 & 2 & 4\\ 0 & 0 & 0 & b \end{bmatrix}$ has det(A) = 100 and trace(A) = 14. The value of |a-b| is

1. 4
2. 3
3. 2
4. 1

4. Let M⁴ = I, (where I denotes the identity matrix) and M ≠ I, M² ≠ I and M³ ≠ I. Then, for any natural number k, M^-1 equals

1. M^4k+1
2. M^4k+2
3. M^4k+3
4. M^4k

5. A real square matrix A is called skew-symmetric if

1. A^T = A
2. A^T = A^-1
3. A^T = -A
4. A^T = A + A^-1

6. For $A=\begin{bmatrix} 1 & \tan x\\ -\tan x & 1 \end{bmatrix}$, the determinant of A^TA^-1 is

1. sec²x
2. cos4x
3. 1
4. 0

7. Perform the following operations on the matrix

\[\begin{bmatrix} 3 & 4 & 45\\ 7 & 9 & 105\\ 13 & 2 & 195 \end{bmatrix}\]

1. Add the third row to the second row.
2. Subtract the third column from the first column.

The determinant of the resultant matrix is

1. 1
2. 2
3. 0
4. 5

8. If any two columns of a determinant $P=\begin{vmatrix} 4 & 7 & 8\\ 3 & 1 & 5\\ 9 & 6 & 2 \end{vmatrix}$ are interchanged, which one of the following statements regarding the value of the determinant is correct?

1. absolute value remains unchanged but sign will change
2. both absolute value and sign will change
3. absolute value will change but sign will not change
4. both absolute value and sign will remain unchanged

9. If the matrix A is such that $A=\begin{bmatrix} 2\\ -4\\ 7 \end{bmatrix} \begin{bmatrix} 1 & 9 & 5 \end{bmatrix}$ then the determinant of A is equal to

1. 48
2. 25
3. 14
4. 0

10. The maximum value of the determinant among all 2 x 2 real symmetric matrices with trace 14 is

1. 25
2. 69
3. 49
4. 38

11. The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is

1. 200
2. 100
3. 150
4. 75

12. Given that the determinant of the matrix $\begin{bmatrix} 1 & 3 & 0\\ 2 & 6 & 4\\ -1 & 0 & 2 \end{bmatrix}$ is -12, the determinant of the matrix $\begin{bmatrix} 2 & 6 & 0\\ 4 & 12 & 8\\ -2 & 0 & 4 \end{bmatrix}$ is

1. -96
2. -24
3. 24
4. 96

13. The determinant of matrix $\begin{bmatrix} 0 & 1 & 2 & 3\\ 1 & 0 & 3 & 0\\ 2 & 3 & 0 & 1\\ 3 & 0 & 1 & 2 \end{bmatrix}$ is

1. 77
2. 66
3. 88
4. 99

14. For matrices of same dimension M, N and scalar c, which one of these properties does not always hold?

1. (M^T)^T = M
2. (cM)^T = c(M)^T
3. (M+N)^T = M^T + N^T
4. MN = NM

15. Which one of the following equations is a correct identity for arbitrary 3 x 3 real matrices P, Q and R?

1. P(Q+R) = PQ + RP
2. (P-Q)² = P² – 2PQ + Q²
3. det(P+Q) = det(P) + det(Q)
4. (P+Q)² = P² + PQ + PQ + Q²