Question

If a matrix \( A \) is symmetric as well as skew-symmetric, then \( A = 0 \).

Hint:

A matrix that is both symmetric and skew-symmetric must be the zero matrix because the only solution to \( A = -A \) is \( A = 0 \).

Answer 1

Step 1: Definition of symmetric matrix. 
A matrix \( A \) is symmetric if \( A^T = A \), i.e., the transpose of \( A \) is equal to \( A \). 
Step 2: Definition of skew-symmetric matrix. 
A matrix \( A \) is skew-symmetric if \( A^T = -A \), i.e., the transpose of \( A \) is equal to the negative of \( A \). 
Step 3: Combining the properties. 
If \( A \) is both symmetric and skew-symmetric, then: \[ A^T = A \quad \text{and} \quad A^T = -A \] Equating the two, we get: \[ A = -A \] This implies that \( 2A = 0 \), or \( A = 0 \). 
Conclusion: 
Thus, the statement is True.