Which of the following is true for $A=\left[\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right]$?

A - B is a diagonal matrix for any value of $\alpha$ if $B=\left[\begin{array}{cc}\alpha & -1 \\ 2 & 5\end{array}\right]$

The correct answer is Option (3) → A - B is a diagonal matrix for any value of $\alpha$ if $B=\left[\begin{array}{cc}\alpha & -1 \\ 2 & 5\end{array}\right]$

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

$\text{(A)}\; A+4I=\begin{bmatrix}5 & -1 \\ 2 & 7\end{bmatrix} \neq (A+4I)^T \Rightarrow \text{False}$

$\text{(B)}\; A-4I=\begin{bmatrix}-3 & -1 \\ 2 & -1\end{bmatrix},\; \text{not skew-symmetric} \Rightarrow \text{False}$

$\text{(C)}\; B=\begin{bmatrix}\alpha & -1 \\ 2 & 5\end{bmatrix}$

$A-B=\begin{bmatrix}1-\alpha & 0 \\ 0 & -2\end{bmatrix} \Rightarrow \text{diagonal for all }\alpha$

$\Rightarrow \text{True}$

$\text{(D)}\; AA^T \text{ is always symmetric, not skew-symmetric} \Rightarrow \text{False}$

The correct statement is (C).