Question

Show that if B is Skeksymmetric n×n matrix where n is an odd positive integer, then B is not invertible

Answer 1

Using the property of the determinant of the matrix, we can say that $detA = detA^T$ , and the definition of the skew-symmetric matrix show us that $- detA^T = det (−A)$ Since det (−A) = (- 1) n⋅detA, we conclude that matrices of odd order have a zero determinant. The matrix with zero determinant is degenerate matrix, and the inversed matrix can exit only for the non-degenerated matrix, Q.E.D.