If a matrix $A = \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$ then which of the following is / are TRUE?

(A) $|A| = 1$

(B) $A$ is a singular matrix.

(C) $-2A = \begin{bmatrix} 10 & -16 \\ -6 & 10 \end{bmatrix}$

(D) $AI = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$ $I$ is an identity matrix of order 2.

Choose the correct answer from the options given below:

Given: $A = \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$

For statement (A): $|A| = 1$

For a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$

$|A| = (5)(5) - (-8)(-3)$

$|A| = 25 - 24$

$|A| = 1$

Statement (A) is TRUE.

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For statement (B): A is a singular matrix

A matrix is singular if its determinant equals 0.

From statement (A), $|A| = 1$.

Since $|A| = 1 \neq 0$, matrix A is not singular.

Statement (B) is FALSE.

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For statement (C): $-2A = \begin{bmatrix} 10 & -16 \\ -6 & 10 \end{bmatrix}$

$-2A = -2 \times \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$

$-2A = \begin{bmatrix} -2 \times 5 & -2 \times (-8) \\ -2 \times (-3) & -2 \times 5 \end{bmatrix}$

$-2A = \begin{bmatrix} -10 & 16 \\ 6 & -10 \end{bmatrix}$

This does not equal $\begin{bmatrix} 10 & -16 \\ -6 & 10 \end{bmatrix}$.

Statement (C) is FALSE.

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For statement (D): $AI = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$

When any matrix is multiplied by an identity matrix, the result is the original matrix.

$AI = A$

$AI = \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$

This does not equal $\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$.

Statement (D) is FALSE.

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Only statement (A) is TRUE.