If A and B are skew-symmetric matrices, then which of the following is not true?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → $A^6+ B^7$ is symmetric
Given: \( A \) and \( B \) are **skew-symmetric matrices**, i.e., \( A^T = -A \) and \( B^T = -B \).
Properties used:
- Odd powers of a skew-symmetric matrix are skew-symmetric: \( (A^{2n+1})^T = -A^{2n+1} \).
- Even powers of a skew-symmetric matrix are symmetric: \( (A^{2n})^T = A^{2n} \).
- Sum of two skew-symmetric matrices is skew-symmetric.
- Sum of a symmetric and a skew-symmetric matrix is neither symmetric nor skew-symmetric in general.
Option-wise check:
\( A^5 + B^7 \): both are odd powers ⇒ both skew-symmetric ⇒ sum is skew-symmetric ⇒ True
\( A^{21} \): odd power ⇒ skew-symmetric ⇒ True
\( B^{18} \): even power ⇒ symmetric ⇒ True
\( A^6 + B^7 \): \( A^6 \) is even ⇒ symmetric, \( B^7 \) is odd ⇒ skew-symmetric ⇒ sum is neither symmetric nor skew-symmetric ⇒ False