If `[(3,5),(7,9)]` is written as A + B, where A is the skew-symmetric matrix and B is the symmetric matrix, then write the matrix A.

Answer

To find the skew-symmetric matrix \( A \) from the given matrix \( P = \begin{pmatrix} 3 & 5 \\ 7 & 9 \end{pmatrix} \), we can follow these steps:

### Step 1: Understand the decomposition of a matrix
Any matrix \( P \) can be expressed as the sum of a symmetric matrix \( B \) and a skew-symmetric matrix \( A \):
\[ P = A + B \]
where \( A \) is skew-symmetric and \( B \) is symmetric.

### Step 2: Use the formulas for symmetric and skew-symmetric matrices
The skew-symmetric matrix \( A \) can be calculated using the formula:
\[ A = \frac{1}{2}(P - P^T) \]
where \( P^T \) is the transpose of matrix \( P \).

### Step 3: Calculate the transpose of \( P \)
First, we need to find the transpose of \( P \):
\[ P^T = \begin{pmatrix} 3 & 5 \\ 7 & 9 \end{pmatrix}^T = \begin{pmatrix} 3 & 7 \\ 5 & 9 \end{pmatrix} \]

### Step 4: Calculate \( P - P^T \)
Now, we subtract \( P^T \) from \( P \):
\[ P - P^T = \begin{pmatrix} 3 & 5 \\ 7 & 9 \end{pmatrix} - \begin{pmatrix} 3 & 7 \\ 5 & 9 \end{pmatrix} = \begin{pmatrix} 3 - 3 & 5 - 7 \\ 7 - 5 & 9 - 9 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix} \]

### Step 5: Calculate \( A \)
Now we can find \( A \) using the formula:
\[ A = \frac{1}{2}(P - P^T) = \frac{1}{2} \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

Thus, the skew-symmetric matrix \( A \) is:
\[ A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]