Find Symmetric Matrix B

Given:

\[ A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \]

Formula:

\[ B = \frac{1}{2}(A + A^T) \]

Step 1: Find A^T

\[ A^T = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} \]

Step 2: Compute A + A^T

\[ A + A^T = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 2 & 6 \end{bmatrix} \]

Step 3: Multiply by 1/2

\[ B = \frac{1}{2} \begin{bmatrix} 2 & 2 \\ 2 & 6 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 3 \end{bmatrix} \]

Final Answer:

\[ \boxed{ B = \begin{bmatrix} 1 & 1 \\ 1 & 3 \end{bmatrix} } \]