Express Matrix as Sum of Symmetric and Skew-Symmetric Matrices
Given:
\[ A = \begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix} \]
Formula Used:
\[ S = \frac{1}{2}(A + A^T), \quad K = \frac{1}{2}(A – A^T) \]
Step 1: Find AT
\[ A^T = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \]
Step 2: Find Symmetric Matrix S
\[ A + A^T = \begin{bmatrix} 6 & 1 & -5 \\ 1 & -4 & -4 \\ -5 & -4 & 4 \end{bmatrix} \]
\[ S = \begin{bmatrix} 3 & \tfrac{1}{2} & -\tfrac{5}{2} \\ \tfrac{1}{2} & -2 & -2 \\ -\tfrac{5}{2} & -2 & 2 \end{bmatrix} \]
Step 3: Find Skew-Symmetric Matrix K
\[ A – A^T = \begin{bmatrix} 0 & -5 & -3 \\ 5 & 0 & -6 \\ 3 & 6 & 0 \end{bmatrix} \]
\[ K = \begin{bmatrix} 0 & -\tfrac{5}{2} & -\tfrac{3}{2} \\ \tfrac{5}{2} & 0 & -3 \\ \tfrac{3}{2} & 3 & 0 \end{bmatrix} \]
Step 4: Verification
\[ S + K = \begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix} = A \]
Conclusion:
The matrix is successfully expressed as the sum of a symmetric and a skew-symmetric matrix.