If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A + A^T$ equals:

Step-by-step Solution:

Given: $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

$$A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$$

$$A + A^T = \begin{pmatrix} 1+1 & 2+3 \\ 3+2 & 4+4 \end{pmatrix} = \begin{pmatrix} 2 & 5 \\ 5 & 8 \end{pmatrix}$$

Note: $A + A^T$ is always a symmetric matrix.

Hence, Option A is correct.