Let A be any skew-symmetric matrix (where $A^T$ is Transpose of matrix A), then which of the following statements are correct?
(A) $A^2$ is a symmetric matrix
(B) $A^2$ is a skew-symmetric matrix
(C) $A^TA = -A^2$
(D) $A^TA-AA^T = 0$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
(A) $A^2$ is a symmetric matrix (Correct)
(B) $A^2$ is a skew-symmetric matrix (Incorrect)
(C) $A^TA = -A^2$ (Correct)
(D) $A^TA-AA^T = 0$ (Correct)
For a skew-symmetric matrix $A$, the following properties hold:
$A^T = -A$
(A) $A^2$ is a symmetric matrix:
$A^2 = A A$
Taking the transpose:
$(A^2)^T = (A A)^T = A^T A^T = (-A) (-A) = A^2$
This confirms that $A^2$ is symmetric.
(B) $A^2$ is a skew-symmetric matrix:
Since $A^2$ is symmetric, this is incorrect.
(C) $A^T A = -A^2$:
$A^T A = (-A) A = -A^2$
This is true.
(D) $A^T A - A A^T = 0$:
$A^T A - A A^T = (-A) A - A (-A) = -A^2 + A^2 = 0$
This is true.