Let A E Mnxn (R). Prove that A has at most Rank(A) +1 distinct eigenvalues. (Hint: It may be helpful to first prove that if v is an eigenvector corresponding to a nonzero eigenvalue then v is in the column space of A.)

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 be a n x n matrix. Suppose that A 7 0nn, A + In, and A? = A. (A) Show that A has exactly 2 eigenvalues: 0, 1. (B) Let v E nullA. Show that v is an eigenvector of A with eigenvalue 0. (C), Let v E colA. Show that v is an eigenvector of A with eigenvalue 1. (D) Show that A is diagonalizable. 3.

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Find the eigenvalues of A, given that A = and its eigenvectors are V₁ = The eigenvalues are and 1 -1 1 4 2 6 12] 7 8 -6 -9 V₂ = -1 and V3 = 0 B

2. Let A E Mnxn(R) be an invertible matrix and let A E R. Prove that A is an eigenvalue for A if and only if is an eigenvalue for A-1. (Note: be sure to address why exists!)

Let 2. 1 -1 D= 0 3 -1 0 0 2 a. Find the eigenvector(s) associated with the eigenvalue X = 2 1 b. Is the vector an eigenvector of D? Why or why not? 3 c. Based on your work so far do you know if D is diagonalizable? Why or why not.

Find the eigenvalues of A=[a b a b] for any a,b of R . Determine all the values of the constants a,b of R for which A is nondefective.