Consider the following. 4 05 A = 6-16 5 04 (a) Compute the characteristic polynomial of A. det(AAI) = 0 (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.) 0 -1 1 0 = -1 has eigenspace span (smallest -value) 0 1 1 6/5 = 9 × has eigenspace span 1 5 6 23 9 has eigenspace span (largest λ-value) 5 (c) Compute the algebraic and geometric multiplicity of each eigenvalue. λ has algebraic multiplicity 2 12 has algebraic multiplicity 1 13 has algebraic multiplicity 1 and geometric multiplicity 2 X and geometric multiplicity -1 and geometric multiplicity 1

Consider the following. A = -4 11 (a) Compute the characteristic polynomial of A. det(A - A1) = (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.) 1, = has eigenspace span (smallest A-value) d2 = has eigenspace span (largest A-value) (c) Compute the algebraic and geometric multiplicity of each eigenvalue. 1 has algebraic multiplicity and geometric multiplicity iz has algebraic multiplicity and geometric multiplicity

Suppose A is a 3 × 3 matrix with real entries that has a complex eigenvalue 3 – 5 with corresponding eigenvector Eigenvalue = Eigenvector = -4+ 8i -1 -6i . Find another eigenvalue and eigenvector for A.

Find the characteristic polynomial cA(x), the eigenvalues of A, and the eigenspaces of A. What are the algebraic and geometric multiplicities of each eigenvalue? Is A diagonalizable?

Consider the following. A = (a) Compute the characteristic polynomial of A. det(A - AI) = (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.) 2₁ = | ^₂ = 4 0 5 6 -1 6 5 04 23 = has eigenspace span has eigenspace span has eigenspace span ↓↑ ↓ 1 1 (c) Compute the algebraic and geometric multiplicity of each eigenvalue. λ, has algebraic multiplicity and geometric multiplicity λ₂ has algebraic multiplicity and geometric multiplicity 23 has algebraic multiplicity and geometric multiplicity (smallest λ-value) (largest λ-value)

Suppose A is a 3 × 3 matrix with real entries that has a complex eigenvalue 9 - 3i with corresponding eigenvector Eigenvalue = Eigenvector = [-5-8i] 1 4i . Find another eigenvalue and eigenvector for A.

Calculate the characteristic polynomial of the following matrix A, and use it to compute eigenvalues of A. 3 -4 -1 A = 3 -20 -16 2 How to enter polynomials. Enter the formula for a polynomial using * to denote multiplication and ^ to denote the exponent. For example, if the answer is the polynomial 3 P (x) = 1 – 2x + -a + 4.5a3 then you should enter 1 - 2*x + (3/7)*x^2 + 4.5*x^3. Do not forget to use * for multiplication! Enter the characteristic polynomial of the matrix A: Note: Use x as the polynomial variable. Enter eigenvalues of A separating them by commas: For example, if you obtain that eigenvalues are 1, -2, and 5 then you should enter 1, -2, 5

Compute the characteristic polynomial of and use it to get the eigenvalues.