If `B=[(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 \( B = P + Q \), where \( B = \begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} \), \( P \) 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 \( P^T = P \), while a skew-symmetric matrix \( Q \) satisfies \( Q^T = -Q \).

### Step 2: Use the formula to separate \( B \) into \( P \) and \( Q \)
We can express any matrix \( B \) as the sum of a symmetric matrix \( P \) and a skew-symmetric matrix \( Q \) using the following formulas:
- \( P = \frac{1}{2}(B + B^T) \)
- \( Q = \frac{1}{2}(B - B^T) \)

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

### Step 4: Calculate \( P \)
Using the formula for \( P \):
\[ P = \frac{1}{2}(B + B^T) = \frac{1}{2}\left(\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix}\right) \]
Now, add the matrices:
\[ B + B^T = \begin{pmatrix} 2 + 2 & 3 + 1 \\ 1 + 3 & 0 + 0 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} \]
Now, divide by 2:
\[ P = \frac{1}{2}\begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]

### Step 5: Final result
Thus, the symmetric matrix \( P \) is:
\[ P = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix} \]