3 Let A = 4 -3 (a) Find the eigenvalues of and the corresponding eigenvectors for the matrix A. (b) Find an matrix P such that t PT AP diagonalizes A.

Let A be a nxn symmetric matrix with real entries. (a) Show that if A is an eigenvector for A, then X must be a real number. (b) If A1 and A2 are two eigenvalues for A with vị and v2 eigenvectors, respectively. Then vi and a2 are orthogonal.

Consider the matrix 2 -4 A = (-²₁ 2¹) -1 (a) Compute the eigenvalues and the corresponding eigenvectors. (b) Let v = = [2, 1], compute A10¹0000 V.

Construct a 3 x 3 symmetric matrix A such that the eigenvalues of A are 1 with multiplicity two and -1 with multiplicity one and where x = [1 1 1 ]T and y = [1 1 -2]T are eigenvectors that correspond to the eigenvalue 1.

For the matrix A, find (if possible) a nonsingular matrix P such that P-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 1 2 4 1 2 P = A = I P-¹AP = N|H m/~ 3 -1 Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal.

Detailed justifications are required!