Which of the following statements are correct?

(A) If A is a square matrix, then $|A^2| = |A|^2$.

(B) If A and B are square matrices of the same order, then det (AB) = det (A) + det (B).

(C) If A is a square matrix of order 3 and $|A|=2$, then the value of $|-3A|$ is 54.

(D) If the matrix $\begin{bmatrix} 5 -x & x -1 \\ 3 &5 \end{bmatrix}$ is singular, then the value of x is 7/2.

Choose the correct answer from the options given below:

Let's check each statement one by one.

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Statement (A): If $A$ is a square matrix, then $|A^2| = |A|^2$

We know that for any two square matrices of the same order, $\det(AB) = \det(A) \cdot \det(B)$

So,

$|A^2| = |A \cdot A| = |A| \cdot |A| = |A|^2$

✅ Statement (A) is correct.

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Statement (B): $\det(AB) = \det(A) + \det(B)$

This is wrong. The correct property is:

$\det(AB) = \det(A) \cdot \det(B)$

It is multiplication, not addition.

❌ Statement (B) is incorrect.

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Statement (C): If $A$ is a square matrix of order 3 and $|A| = 2$, then $|-3A| = 54$

When you multiply a matrix of order $n$ by a scalar $k$, each row gets multiplied by $k$. So the determinant gets multiplied by $k^n$:

$|kA| = k^n \cdot |A|$

Here, $k = -3$, $n = 3$, $|A| = 2$:

$|-3A| = (-3)^3 \cdot |A|$

$= -27 \times 2$

$= -54$

The statement claims the value is $54$ (positive), but the actual value is $-54$.

❌ Statement (C) is incorrect.

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Statement (D): If the matrix $\begin{bmatrix} 5-x & x-1 \\ 3 & 5 \end{bmatrix}$ is singular, then $x = \dfrac{7}{2}$

A singular matrix has its determinant equal to zero.

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant $= ad - bc$

So,

$(5 - x)(5) - (x - 1)(3) = 0$

$25 - 5x - 3x + 3 = 0$

$28 - 8x = 0$

$x = \dfrac{28}{8} = \dfrac{7}{2}$

✅ Statement (D) is correct.

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Statements (A) and (D) are correct.