📘 Question
If
\[
A =
\begin{bmatrix}
2 & 0 & -3 \\
4 & 3 & 1 \\
-5 & 7 & 2
\end{bmatrix}
\]
is expressed as the sum of a symmetric and skew-symmetric matrix, find the symmetric matrix.
✏️ Step-by-Step Solution
Step 1: Use formula
\[
S = \frac{A + A^T}{2}
\]
Step 2: Find transpose
\[
A^T =
\begin{bmatrix}
2 & 4 & -5 \\
0 & 3 & 7 \\
-3 & 1 & 2
\end{bmatrix}
\]
Step 3: Add \(A + A^T\)
\[
A + A^T =
\begin{bmatrix}
4 & 4 & -8 \\
4 & 6 & 8 \\
-8 & 8 & 4
\end{bmatrix}
\]
Step 4: Divide by 2
\[
S =
\begin{bmatrix}
2 & 2 & -4 \\
2 & 3 & 4 \\
-4 & 4 & 2
\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
2 & 2 & -4 \\
2 & 3 & 4 \\
-4 & 4 & 2
\end{bmatrix}
}
\]
Correct Option: (a)
💡 Key Concept
Any matrix can be written as:
- \(S = \frac{A + A^T}{2}\) (symmetric)
- \(K = \frac{A – A^T}{2}\) (skew-symmetric)