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:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) only
The determinant, the multiple \(-2A\) and the product AI are found directly:
\(|A| = 5 \times 5 - (-8)(-3)\)
\(|A| = 25 - 24 = 1\)
\( \displaystyle -2A = \begin{bmatrix}-10&16\\6&-10\end{bmatrix} \)
\( \displaystyle AI = A = \begin{bmatrix}5&-8\\-3&5\end{bmatrix} \)
So |A| = 1, A is non-singular, \(-2A\) has entries −10, 16, 6, −10, and AI = A.
Checking the statements
(A) \(|A| = 1\) — Correct. \(25 - 24 = 1\).
(B) A is a singular matrix. — Incorrect. \(|A| = 1 \ne 0\), so A is non-singular.
(C) \(-2A=\begin{bmatrix}10&-16\\-6&10\end{bmatrix}\) — Incorrect. Every sign is reversed; \(-2A\) has entries −10, 16, 6, −10.
(D) \(AI =\begin{bmatrix}5&0\\0&5\end{bmatrix}\), I is an identity matrix of order 2. — Incorrect. AI = A, whose off-diagonal entries are −8 and −3, not 0.