Question 6:

For an invertible matrix A, prove that A and AI 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 = 1x for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the followin Ax = 1x AX = 1X Ax = 1x AXA -1 = 1xA-1 A/(AX) = A/(1x) Ax = 1x OXAA -1 = 14-1x (4/A)x = (A/1)x A AX = A-1/x Ax/A = 1x/A O XI = 1A -1x IX = 1A-1x (4/A)x = AXA-1 IX = (A/1)x IX = 1xA-1 X = 1A -1x X = 1A -1x x = 1A-1x X = 1xA -1 A-lx = 1x A lx = 1x Alx = 1x A-lx = 1x This shows that |---Select--- v | is an eigenvector of A with eigenvalue --Select-- vFind the dwaracteristic equation and the eigenvalues (and a basis For each of the corresponding elgenspaces) of the matrix. [-3 '1'] (a) the characteristic equation (b) the eigenvalues (Enter your answers From smallest to largest.) W(:) a basis For each of the corresponding eigenspaces