For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1? Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax Ax = Ax AxA-1 = AxA-1 = AA-'x XI = AA-'x x = AA-1x Ax/A = ix/A A/(Ax) = A/(1x) O(A/A)x = (A/2)x Ix = (A/1)x x = AA-1x A-'Ax = A-Ax (A/A)x = ixA-1 Ix = AxA-1 x = ixA-1 OXAA-1 Ix = 1A-1x x = JA-1x A-1x = 1x A-x = 1x A-1x = 1x A-1x = 1x This shows that ---Select-- v is an eigenvector of A- with eigenvalue ---Select- V --Select--- Need Help? 1/x 1/2

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