Show how this matrix is singular if its size is odd arbitrar

Show how this matrix is singular if its size is odd (arbitrary proof)

[linear algebra]

A matrix is said to be skew symmetric if A = -A. For example A = [0 1 -1 0] is skew symmetric, since A = [0 -1 1 0] = -A If A is an arbitrary n times n skew-symmetric matrix and n Is odd. show that .4 must be singular.

Solution

We know that det (A^T) = det (A). If A is skew symmetric, then A^T = -A so that det(A^T) = -det(A). Thus, if A is skew symmetric, then det(A) = - det(A) or det(A) + det(A) = 0 or, 2 (detA) = 0. Then det(A) must be 0. Hence A is singular.