Exercise I (6 x 5 = 30 points) 1. If B = MAM, why is det B = det A? 2. If the entries of A and A" are all positive integers, What are all the possible values of det A and det A-1? 3. Consider the matrices A = all a12 and C = -012 . Let's call A and C sisters. Prove that: Q21 122 -021 a11 (a) Sisters have the same determinant (b) CA = AC = (detA)I, where I is the 2 x 2 identity matrix. (c) The sister of A" is CT. (d) If A is invertible, find the inverse of A in terms of its determinant and its sister. Exercise II (2 x 10 = 20 points) 0 1. Compute the eigenvector of -3 corresponding to eigenvalue -3. 0 0 2. Find the third row of the matrix 1 , so that its characteristic polynomial is -13 + 412 + 51 + 6. b Exercise III (5 x 5 = 25 points) Consider the matrix A = [0.4 0.3] 0.2 0.5 1. Compute the eigenvalues of A. 2. Compute the eigenvectors associated with each of the eigenvalues in A. 3. Use your results in 1. and 2. to diagonalize A. 4. Find a formula for A*. 5. Use your result in 4. to find compute the matrix A'. What does the matrix A approach as k - c? Exercise IV (10 points) Show that the matrices A and B below are similar by finding M so that B = M-'AM: 4= (3 4 and B = 2 3 ] Exercise V (15 points) By following the steps below, diagonalize the matrix A = -2 6 2 into QAQ, where Q is orthogonal: 1. Verify that the eigenvalues of A are 7 and -2. 2. Compute the eigenvectors associated with the given eigenvalues. 3. Make the eigenvectors orthogonal using Gram-Schmidt, and normalize them (make them unit vectors). 4. Store the 3 eigenvectors in a 3 x 3 matrix @, and the eigenvalues in the diagonal matrix A. Please do not rush this exercise. Compute the eigenvectors very carefully and, eventually, make sure that QAQT multiplies back to A. You can ask questions on Campuswire if you need clarification