The value of $2y - 3x$, if $2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix} + \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}$ is :

Let $A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix}$, where $0 \leq \theta \leq 2\pi$. Then which of the following are true?

(A) $|A| = 2 + 2 \sin^{2} \theta$

(B) $|A| = 2 + \sin^{2} \theta$

(C) minimum value of $|A|$ is $1$

(D) maximum value of $|A|$ is $4$

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If a matrix has 12 elements, then the possible orders it can have, are

(A) 1 × 12

(B) 4 × 3

(C) 6 × 3

(D) 6 × 2

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If $\begin{bmatrix} x+2y & -y \\ 3x & 4 \end{bmatrix} = \begin{bmatrix} -4 & 3 \\ 6 & 4 \end{bmatrix}$, then $3x+2y$ is equal to

If the system of equation

$\lambda x + 3y - z = 4$

$x + 2y + z = 20$

$-\lambda x + y + 2z = -5$

does not possess a unique solution, then

Which of the following are correct?

(A) For a square matrix A, if A³ = I, then A⁻¹ = A².

(B) The determinant of only a square matrix can be defined.

(C) If A is a square matrix of order 3, then the number of minors of the matrix A is 3.

(D) If A and B are two non-singular matrices of the same order, then (AB)⁻¹ = B⁻¹A⁻¹.

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If A is a skew-symmetric matrix, then which of the following statements is NOT true?

(A) A is singular if order of A is odd

(B) A is non-singular

(C) $A^{2025}$ is a skew-symmetric matrix

(D) $A^{2025}$ is a symmetric matrix

(E) all diagonal elements of A are zeros

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