Let A and B be n × n matrices. Show that If λ is a nonzero eigenvalue of AB, then it is also an eigenvalue of BA.

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 11 and 2. 6 4-2 4 6 -2 -2 -2 3 Enter the matrices P and D below. (Use a comma to separate matrices as needed. Type exact answers, using radicals as needed. Do not label the matrices.)

Let A = 3 1 2 4 Find the eigenvalues and the corresponding eigenvectors of matrix A ck that it is diagonalizable by computing AP and PD.

Let A be a nxn symmetric matrix with real entries. (a) Show that if A is an eigenvector for A, then X must be a real number. (b) If A1 and A2 are two eigenvalues for A with vị and v2 eigenvectors, respectively. Then vi and a2 are orthogonal.

Let A be an nxn matrix. Determine whether the statement below is true or false. Justify the answer. The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A. The statement is false. The multiplicity of a root r of the characteristic equation of A is the number of eigenvectors corresponding to that root. The statement is true. It is the definition of the multiplicity of a root of the characteristic equation of A. The statement is true. It is the definition of the algebraic multiplicity of an eigenvalue of A. OThe statement is false. The multiplicity of a rootr of the characteristic equation. of A is called the geometric multiplicity of r as an eigenvalue of A.