Express the following as the sum of symmetric and skew - symmetric matrices :
`[(2,-4,5),(1,8,-2),(7,3,9)]`

Answer

To express the matrix \( A = \begin{pmatrix} 2 & -4 & 5 \\ 1 & 8 & -2 \\ 7 & 3 & 9 \end{pmatrix} \) as the sum of a symmetric matrix \( P \) and a skew-symmetric matrix \( Q \), we can use the following formulas:

1. The symmetric matrix \( P \) is given by:
\[ P = \frac{1}{2}(A + A^T) \]

2. The skew-symmetric matrix \( Q \) is given by:
\[ Q = \frac{1}{2}(A - A^T) \]

### Step 1: Calculate the transpose of matrix \( A \)

The transpose of matrix \( A \) is obtained by swapping rows with columns:
\[ A^T = \begin{pmatrix} 2 & 1 & 7 \\ -4 & 8 & 3 \\ 5 & -2 & 9 \end{pmatrix} \]

### Step 2: Calculate the symmetric matrix \( P \)

Now, substitute \( A \) and \( A^T \) into the formula for \( P \):
\[ P = \frac{1}{2} \left( A + A^T \right) = \frac{1}{2} \left( \begin{pmatrix} 2 & -4 & 5 \\ 1 & 8 & -2 \\ 7 & 3 & 9 \end{pmatrix} + \begin{pmatrix} 2 & 1 & 7 \\ -4 & 8 & 3 \\ 5 & -2 & 9 \end{pmatrix} \right) \]

Calculating \( A + A^T \):
\[ A + A^T = \begin{pmatrix} 2 + 2 & -4 + 1 & 5 + 7 \\ 1 - 4 & 8 + 8 & -2 + 3 \\ 7 + 5 & 3 - 2 & 9 + 9 \end{pmatrix} = \begin{pmatrix} 4 & -3 & 12 \\ -3 & 16 & 1 \\ 12 & 1 & 18 \end{pmatrix} \]

Now, divide by 2:
\[ P = \frac{1}{2} \begin{pmatrix} 4 & -3 & 12 \\ -3 & 16 & 1 \\ 12 & 1 & 18 \end{pmatrix} = \begin{pmatrix} 2 & -\frac{3}{2} & 6 \\ -\frac{3}{2} & 8 & \frac{1}{2} \\ 6 & \frac{1}{2} & 9 \end{pmatrix} \]

### Step 3: Calculate the skew-symmetric matrix \( Q \)

Now, substitute \( A \) and \( A^T \) into the formula for \( Q \):
\[ Q = \frac{1}{2} \left( A - A^T \right) = \frac{1}{2} \left( \begin{pmatrix} 2 & -4 & 5 \\ 1 & 8 & -2 \\ 7 & 3 & 9 \end{pmatrix} - \begin{pmatrix} 2 & 1 & 7 \\ -4 & 8 & 3 \\ 5 & -2 & 9 \end{pmatrix} \right) \]

Calculating \( A - A^T \):
\[ A - A^T = \begin{pmatrix} 2 - 2 & -4 - 1 & 5 - 7 \\ 1 + 4 & 8 - 8 & -2 - 3 \\ 7 - 5 & 3 + 2 & 9 - 9 \end{pmatrix} = \begin{pmatrix} 0 & -5 & -2 \\ 5 & 0 & -5 \\ 2 & 5 & 0 \end{pmatrix} \]

Now, divide by 2:
\[ Q = \frac{1}{2} \begin{pmatrix} 0 & -5 & -2 \\ 5 & 0 & -5 \\ 2 & 5 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -\frac{5}{2} & -1 \\ \frac{5}{2} & 0 & -\frac{5}{2} \\ 1 & \frac{5}{2} & 0 \end{pmatrix} \]

### Final Result

Thus, we can express the matrix \( A \) as:
\[ A = P + Q \]
Where:
\[ P = \begin{pmatrix} 2 & -\frac{3}{2} & 6 \\ -\frac{3}{2} & 8 & \frac{1}{2} \\ 6 & \frac{1}{2} & 9 \end{pmatrix} \quad \text{(symmetric)} \]
\[ Q = \begin{pmatrix} 0 & -\frac{5}{2} & -1 \\ \frac{5}{2} & 0 & -\frac{5}{2} \\ 1 & \frac{5}{2} & 0 \end{pmatrix} \quad \text{(skew-symmetric)} \]