If `A=[(2,3),(1,0)]` =P +Q where P is symmetric matrix and Q is skew -skew-symmetric matrix then the value of matrix P is :

Options

A. `[(0,-2),(2,0)]`
B. `[(2,2),(2,0)]`
C. `[(2,-2),(-2,0)]`
D. `[(1,2),(0,1)]`

Answer

To find the symmetric matrix \( P \) from the equation \( A = P + Q \), where \( A = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} \), \( P \) is symmetric, and \( Q \) is skew-symmetric, we can follow these steps:

### Step-by-Step Solution

1. **Understand the properties of symmetric and skew-symmetric matrices**:
- A matrix \( P \) is symmetric if \( P^T = P \).
- A matrix \( Q \) is skew-symmetric if \( Q^T = -Q \).

2. **Write the equation**:
\[ A = P + Q \]
where \( A = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} \).

3. **Take the transpose of both sides**:
\[ A^T = P^T + Q^T \]
Since \( P \) is symmetric, \( P^T = P \), and since \( Q \) is skew-symmetric, \( Q^T = -Q \). Therefore, we can rewrite the equation as:
\[ A^T = P - Q \]

4. **Calculate \( A^T \)**:
\[ A^T = \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} \]

5. **Now we have two equations**:
\[ A = P + Q \quad (1) \]
\[ A^T = P - Q \quad (2) \]

6. **Add equations (1) and (2)**:
\[ (P + Q) + (P - Q) = A + A^T \]
This simplifies to:
\[ 2P = A + A^T \]

7. **Calculate \( A + A^T \)**:
\[ A + A^T = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} \]

8. **Substitute back to find \( P \)**:
\[ 2P = \begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} \]
Divide both sides by 2:
\[ P = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]

### Final Result
The value of matrix \( P \) is:
\[ P = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]