Consider the general 2 x 2 real symmetric matrix a b Prove that A has an eigenvalue of mul- A: tiplicity two if and only if it is a scalar matrix (that is, a matrix of the form r12, where r is a constant).

The matrix A is a real symmetric 5 x 5 matrix with eigenvalues 3, 7, 10, 15, and 21. The vector xị is an eigenvector of A with eigenvalue 3, the vector x2 is an eigenvector of eigenvalue 10, the vector x4 is an eigenvector of A with eigenvalue 15, and the vector x; is an eigenvector of A with eigenvalue 21. with eigenvalue 7, the vector x3 is an eigenvector of A with Compute the dot products of the eigenvectors a; with each other. 21. x2 = I1. X3 = X1. 24 = x2 · X3 = I2 · X4 = X2 · X5 = 13 · X4 = 13 · X5 = X4: x5 =

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.

Consider the matrix 2 2 A = 4 2 2 (a) Compute the eigenvalues and eigenvectors of A. (b) Find rank one matrices B and C such that A = B + C and BC 0, explaining why the matrices you found have these properties. (hint: use your answer to part (a))