1. **State the problem:** We are given the matrix $$\begin{bmatrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{bmatrix}$$ and need to express it as a sum of a symmetric matrix and a skew-symmetric matrix. 2. **Recall definitions:** - A symmetric matrix $S$ satisfies $S = S^T$. - A skew-symmetric matrix $K$ satisfies $K = -K^T$. 3. **Formula:** Any square matrix $A$ can be decomposed as $$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$$ where - $S = \frac{A + A^T}{2}$ is symmetric, - $K = \frac{A - A^T}{2}$ is skew-symmetric. 4. **Calculate $A^T$:** $$A^T = \begin{bmatrix}7 & -1 & -1 \\ -3 & 1 & 0 \\ -3 & 0 & 1\end{bmatrix}$$ 5. **Calculate symmetric part $S$:** $$S = \frac{1}{2}(A + A^T) = \frac{1}{2} \begin{bmatrix}7+7 & -3-1 & -3-1 \\ -1-3 & 1+1 & 0+0 \\ -1-3 & 0+0 & 1+1\end{bmatrix} = \frac{1}{2} \begin{bmatrix}14 & -4 & -4 \\ -4 & 2 & 0 \\ -4 & 0 & 2\end{bmatrix} = \begin{bmatrix}7 & -2 & -2 \\ -2 & 1 & 0 \\ -2 & 0 & 1\end{bmatrix}$$ 6. **Calculate skew-symmetric part $K$:** $$K = \frac{1}{2}(A - A^T) = \frac{1}{2} \begin{bmatrix}7-7 & -3+1 & -3+1 \\ -1+3 & 1-1 & 0-0 \\ -1+3 & 0-0 & 1-1\end{bmatrix} = \frac{1}{2} \begin{bmatrix}0 & -2 & -2 \\ 2 & 0 & 0 \\ 2 & 0 & 0\end{bmatrix} = \begin{bmatrix}0 & -1 & -1 \\ 1 & 0 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ 7. **Verify:** $$S + K = \begin{bmatrix}7 & -2 & -2 \\ -2 & 1 & 0 \\ -2 & 0 & 1\end{bmatrix} + \begin{bmatrix}0 & -1 & -1 \\ 1 & 0 & 0 \\ 1 & 0 & 0\end{bmatrix} = \begin{bmatrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{bmatrix} = A$$ **Final answer:** $$A = \begin{bmatrix}7 & -2 & -2 \\ -2 & 1 & 0 \\ -2 & 0 & 1\end{bmatrix} + \begin{bmatrix}0 & -1 & -1 \\ 1 & 0 & 0 \\ 1 & 0 & 0\end{bmatrix}$$