DETERMINANTS AND EIGENVALUES 1. Introduction Gauss ...

4. EIGENVALUES AND EIGENVECTORS FOR n × n MATRICES 99Exercises for Section 4.1. Find the eigenvalues and eigenvectors for each of the following matrices. Usethe method given in the text for solving the characteristic equation if it has degreegreater[ than two. ]5 −3(a) .2 0⎡⎤3 −2 −2(b) ⎣ 0 0 1⎦.1 0 −1⎡⎤2 −1 −1(c) ⎣ 0 0 −2⎦.0 1 3⎡⎤4 −1 −1(d) ⎣ 0 2 −1⎦.1 0 32. You are a mechanical engineer checking for metal fatigue in a vibrating system.Mathematical ⎡ analysis reduces the problem to finding eigenvectors for the matrixA = ⎣ −2 1 0⎤⎡1 −2 1⎦. A member of your design team tells you that v = ⎣ 1 ⎤1 ⎦ is0 1 −21an eigenvector for A. What is the quickest way for you to check if this is correct?3. As in⎡the previous problem, some other member of your design team tells youthat v = ⎣ 0 ⎤0 ⎦ is a basis for the eigenspace of the same matrix corresponding to0one of its eigenvalues. What do you say in return?4. Under what circumstances can zero be an eigenvalue of the square matrixA? Could A be non-singular in this case? Hint: The characteristic equation isdet(A − λI) =0.5. Let A be a square matrix, and suppose λ is an eigenvalue for A with eigenvectorv. Show that λ 2 is an eigenvalue for A 2 with eigenvector v. What about λ n andA n for n a positive integer?6. Suppose A is non-singular. Show that λ is an eigenvalue of A if and only ifλ −1 is an eigenvalue of A −1 . Hint. Use the same eigenvector.7. (a) Show that det(A−λI) is a quadratic polynomial in λ if A isa2×2 matrix.(b) Show that det(A − λI) is a cubic polynomial in λ if A isa3×3 matrix.(c) What would you guess is the coefficient of λ n in det(A − λI) for A an n × nmatrix?8. (Optional) Let A be an n × n matrix with entries not involving λ. Prove ingeneral that det(A − λI) is a polynomial in λ of degree n. Hint: Assume B(λ) isann×nmatrix such that each column has at most one term involving λ and that term