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

Options

A. `[(5,5),(1,5)]`
B. `[(2,2),(2,0)]`
C. `[(-1,0),(1,2)]`
D. `[(1,0),(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 1: Understand the properties of symmetric and skew-symmetric matrices
- A symmetric matrix \( P \) satisfies \( P^T = P \).
- A skew-symmetric matrix \( Q \) satisfies \( Q^T = -Q \).

### Step 2: Write the equations for \( A \)
Given \( A = P + Q \), we can also write:
\[ A^T = P^T + Q^T \]

### Step 3: Transpose the matrix \( A \)
Calculate the transpose of \( A \):
\[ A^T = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix}^T = \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} \]

### Step 4: Substitute the properties into the equation
Using the properties of \( P \) and \( Q \):
\[ A^T = P + (-Q) \implies A^T = P - Q \]

### Step 5: Set up the equations
Now we have two equations:
1. \( A = P + Q \)
2. \( A^T = P - Q \)

Substituting the values of \( A \) and \( A^T \):
1. \( \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} = P + Q \)
2. \( \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} = P - Q \)

### Step 6: Add the two equations
Adding the two equations:
\[ \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} = (P + Q) + (P - Q) \implies 2P = \begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} \]

### Step 7: Solve for \( P \)
Dividing both sides by 2:
\[ P = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]

### Final Answer
Thus, the symmetric matrix \( P \) is:
\[ P = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]