📘 Question
The matrix
\[
\begin{bmatrix}
0 & 5 & -7 \\
-5 & 0 & 11 \\
7 & -11 & 0
\end{bmatrix}
\]
is:
(a) a skew-symmetric matrix
(b) a symmetric matrix
(c) a diagonal matrix
(d) an upper triangular matrix
✏️ Step-by-Step Solution
Step 1: Recall definition
A matrix is skew-symmetric if:
\[
A^T = -A
\]
Step 2: Check elements
- \(a_{12} = 5\), \(a_{21} = -5\)
- \(a_{13} = -7\), \(a_{31} = 7\)
- \(a_{23} = 11\), \(a_{32} = -11\)
- Diagonal elements = 0
All conditions satisfy:
\[
a_{ij} = -a_{ji}
\]
Step 3: Conclusion
Hence, the matrix is skew-symmetric.
✅ Final Answer
\[
\boxed{(a)\; \text{skew-symmetric matrix}}
\]
💡 Key Concept
In a skew-symmetric matrix:
- Diagonal elements are always zero
- Elements are negatives across diagonal