Question #24672

1 a) Show that if A is nonsingular symmetric matrix, then A^-1 is also symmetric. Write your
justication 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.

Expert's answer

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)

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)

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