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:

Given $A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$

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Checking Statement (A): A is non-singular matrix

A matrix is non-singular if its determinant ≠ 0.

For a 2×2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, determinant = ad - bc

$\text{det}(A) = (1)(3) - (2)(-2)$

$= 3 + 4$

$= 7$

Since det(A) = 7 ≠ 0, A is non-singular.

Statement (A) is TRUE.

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Checking Statement (B): $A^T = A$

The transpose of A is:

$A^T = \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}$

Comparing with $A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$:

$A^T ≠ A$

Statement (B) is FALSE.

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Checking Statement (C): A is not invertible matrix

A matrix is invertible if and only if det(A) ≠ 0.

From Statement (A), det(A) = 7 ≠ 0.

Therefore, A is invertible.

Statement (C) is FALSE.

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Checking Statement (D): A is not skew-symmetric matrix

A matrix is skew-symmetric if $A^T = -A$.

$-A = \begin{bmatrix} -1 & -2 \\ 2 & -3 \end{bmatrix}$

$A^T = \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}$

Since $A^T ≠ -A$, A is not skew-symmetric.

Statement (D) is TRUE.

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Statements (A) and (D) are TRUE.