📘 Question
If \[ A = \begin{bmatrix} 3 & 5 \\ 7 & 9 \end{bmatrix} \] is written as \(A = P + Q\), where \(P\) is symmetric and \(Q\) is skew-symmetric, find matrix \(P\).
✏️ Step-by-Step Solution
Step 1: Use decomposition formula
For any matrix:
\[
P = \frac{A + A^T}{2}
\]
Step 2: Find transpose \(A^T\)
\[
A^T =
\begin{bmatrix}
3 & 7 \\
5 & 9
\end{bmatrix}
\]
Step 3: Add \(A + A^T\)
\[
A + A^T =
\begin{bmatrix}
3+3 & 5+7 \\
7+5 & 9+9
\end{bmatrix}
=
\begin{bmatrix}
6 & 12 \\
12 & 18
\end{bmatrix}
\]
Step 4: Divide by 2
\[
P =
\frac{1}{2}
\begin{bmatrix}
6 & 12 \\
12 & 18
\end{bmatrix}
=
\begin{bmatrix}
3 & 6 \\
6 & 9
\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
3 & 6 \\
6 & 9
\end{bmatrix}
}
\]
💡 Key Concept
Any square matrix can be written as:
- Symmetric part: \(P = \frac{A + A^T}{2}\)
- Skew-symmetric part: \(Q = \frac{A – A^T}{2}\)
This decomposition is always possible.