Symmetric Matrix and Proof that A + A^T is Symmetric

Definition:

A square matrix \(A\) is called symmetric if \[ A^T = A \]

Given:

\[ A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix} \]

Step 1: Find A^T

\[ A^T = \begin{bmatrix} 2 & 5 \\ 4 & 6 \end{bmatrix} \]

Step 2: Compute A + A^T

\[ A + A^T = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix} + \begin{bmatrix} 2 & 5 \\ 4 & 6 \end{bmatrix} = \begin{bmatrix} 4 & 9 \\ 9 & 12 \end{bmatrix} \]

Step 3: Find Transpose of Result

\[ (A + A^T)^T = \begin{bmatrix} 4 & 9 \\ 9 & 12 \end{bmatrix} \]

Conclusion:

\[ (A + A^T)^T = A + A^T \]

Hence, A + A^T is a symmetric matrix.