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

Options

A. `[(2,3),(1,0)]`
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 \( A \) is given as:

\[ A = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} \]

and \( Q \) is a skew-symmetric matrix, we can use the property that the symmetric part \( P \) of a matrix \( A \) can be calculated using the formula:

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

### Step-by-Step Solution:

1. **Calculate the Transpose of Matrix A**:
\[ A^T = \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} \]

2. **Add Matrix A and its Transpose**:
\[ A + A^T = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 2 + 2 & 3 + 1 \\ 1 + 3 & 0 + 0 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} \]

3. **Multiply the Result by \( \frac{1}{2} \)**:
\[ P = \frac{1}{2} \begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} \frac{4}{2} & \frac{4}{2} \\ \frac{4}{2} & \frac{0}{2} \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]

Thus, the value of the symmetric matrix \( P \) is:

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