Linear Algebra Questions

7. Consider the matrix A = [4 8 0 2 4 0 0 0 10] Check that the vectors bl = (2,1,0), 112 = (1,2,0), and D3 = (0,0,1) are eigenvectors of A by computing the products A131, 1413;, and Abg and comparing the results to b1, b2, and b3. From this comparison, what are the eigenvalues for each of these eigenvectors? The set B = {b1, b2, b3} is a basis of R3. (The matrixA just acts by scaling these directions.) Now consider the vector 12 which has coordinates (17);, = (2,3, 1) relative to this basis of eigenvectors. Determine (A103 by using this scaling fact and then use these coordinates to nd the result of Av. 2 1 O This matrix can be diagonalized using P = 1 2 O] (which has inverse given by 0 0 1 2/5 1/5 0 P1 = l 1/5 2/5 0]). Give the diagonal matrix!) such that P'iAP = D. (Note: You 0 0 1 shouldn't need to compute this product to find D I) Use the diagonalization ofA to compute A5. (You will need P'1 and to do some matrix multiplication to do thisl)