9.M.3 A 2 × 2 matrix A is symmetric, and has eigenvalues 3 and -2. A 3-eigenvector is [3]. 5 Find A. Hint: Because A is symmetric, you know that every -2-eigenvector is perpendicular to every 3-eigenvector, which lets you find one. Then use the diagonalization theorem to describe A as PDP-¹.

Let B be a 2 × 2 matrix such that 7B²5B+3I = 0. Is B invertible? It so, what are the eigenvalues of B-¹? Justify your answer.

2 Consider the matrix A = -8 12 A. Calculate both eigenvalues of the matrix. B. Choose only one of the two eigenvalues (does not matter which one), and determine a corresponding eigenvector.

7. Determine all 2 × 2 matrices A with eigenvalues 1 and –1 and corresponding eigenvectors and respectively.

Suppose that is an eigenvector for both the matrix A and the matrix B, with corresponding eigenvalue X for A and corresponding eigenvalue μ for B. Show that is an eigenvector of A + B and AB. Proof: Since Av = A and B = μv: (i) (A + B) = and (ii) ABv= A. Au + Bu B. A(Bu) = A(μv) C. λύ + μύ D. (X +μ) E. λμῦ F. μ(Aΰ) = μ(λύ) Q.E.D.

You need to check your QR factorization code using a simple matrix. Use the characteristic equation to solve for the eigenvalues by hand. 3 [9] -1

Please answer all parts.