Find Symmetric and Skew-Symmetric Matrices X and Y

Given:

\[ A = \begin{bmatrix} 3 & 2 & 7 \\ 1 & 4 & 3 \\ -2 & 5 & 8 \end{bmatrix} \]

Formula Used:

\[ X = \frac{1}{2}(A + A^T), \quad Y = \frac{1}{2}(A – A^T) \]

Step 1: Find A^T

\[ A^T = \begin{bmatrix} 3 & 1 & -2 \\ 2 & 4 & 5 \\ 7 & 3 & 8 \end{bmatrix} \]

Step 2: Find X = (A + A^T)/2

\[ A + A^T = \begin{bmatrix} 6 & 3 & 5 \\ 3 & 8 & 8 \\ 5 & 8 & 16 \end{bmatrix} \]

\[ X = \begin{bmatrix} 3 & \tfrac{3}{2} & \tfrac{5}{2} \\ \tfrac{3}{2} & 4 & 4 \\ \tfrac{5}{2} & 4 & 8 \end{bmatrix} \]

Step 3: Find Y = (A − A^T)/2

\[ A – A^T = \begin{bmatrix} 0 & 1 & 9 \\ -1 & 0 & -2 \\ -9 & 2 & 0 \end{bmatrix} \]

\[ Y = \begin{bmatrix} 0 & \tfrac{1}{2} & \tfrac{9}{2} \\ -\tfrac{1}{2} & 0 & -1 \\ -\tfrac{9}{2} & 1 & 0 \end{bmatrix} \]

Conclusion:

\[ A = X + Y \]

Where X is symmetric and Y is skew-symmetric.