(4) Prove that if A is an eigenvalue of an invertible matrix A and x is a corresponding eigenvec- tor, then 1/A is an eigenvalue of A- and x is a corresponding eigenvector.

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Let M be a 2 x 2 matrix with eigenvalues A1 = -0.9, A2 =-0.75 with corresponding eigenvectors V1 V2 Consider the difference equation Mx: with initial condition xo Write the initial condition as a linear combination of the eigenvectors of M. That is, write xo = ¢¡V1 +C2V2 Vi+ V2 In general, X Vi+ )* v2 Specifically, x2 For large k, Xk →

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.

Let A = = -8 2 2 2 2 -5 4 4 (1) Compute the characteristic polynomial of the matrix A. (2) Find the eigenvalue of A and their multiplicities. (3) Is A invertible? Why?

Suppose v is an eigenvector of a matrix A with corresponding eigenvalue A, and suppose c is a (fixed) scalar. Show based on the definition of eigenvector and eigenvalue that v is an eigenvector of cI – A with corresponding eigenvalue c – A.

(3) Let A be an n xn matrix such that A? = A. Prove that A must has 0 or 1 as an eigenvalue.