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 $\text{det (AB) = det (A) + det (B)}$.
(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:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
Check each statement:
(A) If $A$ is square, $|A^2| = |A|^2 ✅$ (True)
(B) For square matrices, $\det(AB) = \det(A) \cdot \det(B) ❌$ (False, not sum)
(C) $A$ is order 3, $|A| = 2$, then $|-3A| = (-3)^3 |A| = -27*2 = -54 ❌$
(D) Matrix $\begin{bmatrix} 5-x & x-1 \\ 3 & 5 \end{bmatrix}$ is singular → determinant = 0
Determinant: $(5-x)(5) - (x-1)(3) = 25 - 5x - 3x + 3 = 28 - 8x = 0$
$x = \frac{28}{8} = \frac{7}{2} ✅$