Express the following as the sum of symmetric and skew - symmetric matrices :
`[(6,-2,2),(-2,3,-1),(2,-1,3)]`

Answer

To express the given matrix \( A = \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} \) as the sum of a symmetric matrix \( P \) and a skew-symmetric matrix \( Q \), we can follow these steps:

### Step 1: Calculate the Transpose of Matrix A
The transpose of matrix \( A \), denoted as \( A^T \), is obtained by swapping the rows and columns of \( A \).

\[ A^T = \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} \]

### Step 2: Find the Symmetric Matrix \( P \)
The symmetric matrix \( P \) can be calculated using the formula:

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

Substituting the values of \( A \) and \( A^T \):

\[ P = \frac{1}{2} \left( \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} + \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} \right) \]

Calculating the sum:

\[ = \frac{1}{2} \begin{pmatrix} 12 & -4 & 4 \\ -4 & 6 & -2 \\ 4 & -2 & 6 \end{pmatrix} \]

Now, divide each element by 2:

\[ P = \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} \]

### Step 3: Find the Skew-Symmetric Matrix \( Q \)
The skew-symmetric matrix \( Q \) can be calculated using the formula:

\[ Q = \frac{1}{2}(A - A^T) \]

Substituting the values of \( A \) and \( A^T \):

\[ Q = \frac{1}{2} \left( \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} - \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} \right) \]

Calculating the difference:

\[ = \frac{1}{2} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

Thus, we have:

\[ Q = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

### Final Result
Therefore, the matrix \( A \) can be expressed as the sum of the symmetric matrix \( P \) and the skew-symmetric matrix \( Q \):

\[ A = P + Q = \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]