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: 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
Given:
$A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$
Determinant:
$|A| = (1)(3) - (2)(-2) = 3 + 4 = 7 \ne 0$
Hence, $A$ is non-singular ⇒ (A) is true, (C) is false.
Check symmetry:
$A^T = \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}$
Since $A^T \ne A$, it is not symmetric.
Check skew-symmetry:
$-A = \begin{bmatrix} -1 & -2 \\ 2 & -3 \end{bmatrix}$
$A^T \ne -A$, hence it is not skew-symmetric.
Therefore, correct statements:
(A) A is non-singular matrix
(D) A is not skew-symmetric matrix