Question

The inverse of a skew-symmetric matrix is

(A) A symmetric matrix if it exists

(B) A skew-symmetric matrix if it exists

(C) Transpose of the original matrix

(D) May not exist

Answer 1

Answer is (B) A skew-symmetric matrix if it exists

If A is a skew-symmetric matrix of odd order, then |A| = 0.

So, inverse does not exist.

Let A be of even order. Then

AA^-1 = A^-1A = I_n ⇒ (AA^-1)^T = (A^-1A)^T = I_n

⇒ (A^-1)^T A^T = A^T(A^-1)^T = I_n

⇒ (A^-1)^T (-A) = (-A)(A^-1)^T = I_n

So, (A^-1)^T = -A^-1 (inverse of a matrix is unique).