Answer to Question #24672 in Linear Algebra for Mikhail Vega

Question

1 a) Show that if A is nonsingular symmetric matrix, then A^-1 is also symmetric. Write your
justication in clear sentences.
b) An n x n matrix A is called skew-symmetric if A = -A^T . Show that if n is odd a skew-symmetric matrix is singular.

Answer 1

1
A^t=A

A^(-1)A=I=AA^(-1)
(A^(-1)A)^t=I^t
A^t& *& (A^(-1))^t = I=AA^(-1)
A& *& (A^(-1))^t =AA^(-1)
(A^(-1))^t =A^(-1)& =>& A^(-1) - symmetric

2
det(t)=det(A^t)=det(-A)=(-1)^(2K+1)*det(A)=det(A)& =>& det(A)=0 =>& there is no A^(-1)