7. (Review) Determine the characteristic polynomial, the eigenvalues, a basis for the eigenspace, and the algebraic and geometric multiplicities of each eigenvalue for the matrix 1 3] -2 6 A =

2. Without using Sage, find • the characteristic polynomial • the eigenvalues (include the multiplicity of each) • bases for the eigenspaces (indicate the dimension of the eigenspaces) of the following matrices: [30 (a) B= (b) C: = 4 (c) D = -2 10 (Once you've found the characteristic polynomial, you may find -20 its factorization by using Sage. For example, to factor x²-3x+2 in Sage: f = x^2-3*x+2 f.factor())

Calculate the characteristic polynomial of the following matrix A, and use it to compute the eigenvalues of A. -8 4 A = 14 4 6 -2 4 -6 How to enter polynomials: something like 2- 3*r + 4*r^2 - 5*rA3. Remember to use * for multiplication! Enter the characteristic polynomial of the matrix A: Note: Use r as the polynomial variable. Enter the eigenvalues of A separating them by commas: something like 1, -2, 5.

Suppose the characteristic equation for a matrix A is given by X3 + 3X2 – 9A – 27 = 0. Find the eigenvalues of the matrix A and give the algebraic multiplicity of each eigenvalue. - Eigenvalue: A = 3 Eigenvalue: A = Algebraic multiplicity: Ex: 5 Algebraic multiplicity: