Let A be a matrix such that \(A = \begin{bmatrix}1&2\\-2&3\end{bmatrix}\). Then which of the following are TRUE?
(A) A is non-singular matrix
(B) \(A^T = A\)
(C) A is not invertible matrix
(D) A is not skew-symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (D) only
\(|A| = (1)(3) - (2)(-2) = 7\)
\( \displaystyle A^T = \begin{bmatrix}1&-2\\2&3\end{bmatrix} \)
A skew-symmetric matrix has every diagonal entry 0, but here the diagonal entries are 1 and 3.
So \(|A| = 7 \ne 0\), \(A^T \ne A\), and A is not skew-symmetric.
Checking the statements
(A) A is non-singular matrix — Correct. \(|A| = 7 \ne 0\).
(B) \(A^T = A\) — Incorrect. \(a_{12} = 2\) but \(a_{21} = -2\).
(C) A is not invertible matrix — Incorrect. \(|A| \ne 0\), so \(A^{-1}\) exists (non-singular means invertible).
(D) A is not skew-symmetric matrix — Correct. \(a_{11} = 1 \ne 0\).