Question

True or false. Linear Algebra.
Each eigenvector of an invertible marix A is also an eigenvector of A^-1

Answer 1

If k is an eigenvalue for the matrix A, then


Ax = kx


A^(-1)*Ax = A^(-1)*(kx)


I*x = A^(-1)*(kx)


x = A^(-1)*(kx)


x = (A^(-1)x)*k


(1/k)*x = A^(-1)x


A^(-1)x = (1/k)*x


So 1/k is the eigenvalue for the matrix A^(-1)


So the statement "Each eigenvector of an invertible marix A is also an eigenvector of A^-1" is false (since k does not equal 1/k)

Notice how this is only true if k = 1 or k = -1