Find bases for the eigenspaces associated with each eigenvalue of the matrix 3 0 0 -2 3 -2 2 0 5 A = Find, if possible, a nonsingular matrix P such that P−¹AP is diagonal.

Let A be a symmetric n x n matrix. Let A1 and A2 be two eigenvalues with eigenvectors vị and v2 for A, respectively. Show that vi and v2 are orthogonal.

Find a 2 x 2 matrix A such that the entries of A are all real numbers and –4 is an eigenvalue of A, or explain why no such matrix exists.

Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.) 0 9 9 9 0 9 9 9 9 d; = For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.) dim(x;) =