1 a) Given the matrix A = 8 5 (i) Find its characteristics equations, and its eigenvalues. (ii) Find the eigenvalues of A4 and A-3 [4] [2] [4] (iii) Verify the Cayley Hamilton's theorem. (iv) Using the Cayley Hamilton's theorem, find the matrix polynomial that determines A-1, and hence find A-. [7] -3 1 3 10 2 -2 [8] b) Find the eigenvalues, and the corresponding eigenvectors and eigenspaces of the matrix B = 20 hence use the results to construct a matrix Q that orthogonally diagonalizes B. c) Give a matrix representation of the quadratic form f(x, y, z) = 4x+9y + 1622-2xy + 4xz-6yz and determine if the quadratic form is positive or negative definite. [5]