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:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
The results used are the product rule \(|AB| = |A|\,|B|\) and, for a matrix of order n, \(|kA| = k^n|A|\).
Checking the statements
(A) If A is a square matrix, then \(|A^2| = |A|^2\). — Correct. \(|A^2| = |AA| = |A|\,|A| = |A|^2\).
(B) If A and B are square matrices of the same order, then \(\det(AB) = \det(A) + \det(B)\). — Incorrect. \(\det(AB) = \det(A)\det(B)\), the product, not the sum.
(C) If A is a square matrix of order 3 and \(|A| = 2\), then the value of \(|-3A|\) is 54. — Incorrect. \(|-3A| = (-3)^3|A| = -27 \times 2 = -54\); the sign of k is kept for odd order.
(D) If the matrix \(\begin{bmatrix}5 - x & x - 1\\ 3 & 5\end{bmatrix}\) is singular, then the value of \(x\) is 7/2. — Correct. A singular matrix has determinant 0:
\( \displaystyle 5(5 - x) - 3(x - 1) = 0 \)
\( \displaystyle 28 - 8x = 0 \)
\( \displaystyle x = \frac{7}{2} \)