If $A^T =\begin{bmatrix}-2&3\\1&2\end{bmatrix}$, and $B =\begin{bmatrix}-1&0\\1&2\end{bmatrix}$, then the matrix $(A + 2B)^T$ is

$\begin{bmatrix}-4&5\\1&6\end{bmatrix}$

The correct answer is Option (3) → $\begin{bmatrix}-4&5\\1&6\end{bmatrix}$

Given:

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

$B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$

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

$A + 2B = \begin{bmatrix} -2 & 1 \\ 3 & 2 \end{bmatrix} + 2\begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$

$A + 2B = \begin{bmatrix} -2-2 & 1+0 \\ 3+2 & 2+4 \end{bmatrix} = \begin{bmatrix} -4 & 1 \\ 5 & 6 \end{bmatrix}$

$(A+2B)^T = \begin{bmatrix} -4 & 5 \\ 1 & 6 \end{bmatrix}$

Answer: ${\begin{bmatrix} -4 & 5 \\ 1 & 6 \end{bmatrix}}$