Consider the matrix 22 1 2 A = 2 -1 2 4 0 a) Diagonalize the matrix in the form A = SAS-1, with S a matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Is the matrix S an orthogonal matrix ? Why / why not? c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) the rank of the matrix A. a. b. the determinant of the matrix A. C. the null space of the matrix A. d) Determine if the matrix B = (A+A7) ¹ is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. (Hint: use the elimination method and Hermite's theorem).