DETERMINANTS AND EIGENVALUES 1. Introduction Gauss ...

7. REVIEW 109⎡3. A = ⎣ 0 1 1⎤1 0 1⎦. Find the eigenvalues and eigenvectors of A.1 1 0⎡⎤1 3 1 1⎢ 1 2 4 3⎥4. Find det ⎣⎦.2 4 1 11 1 6 15. Each of the following statements is not generally true. In each case explainbriefly why it is false.(a) An n × n matrix A is invertible if and only if det A =0.(b) If A is an n × n real matrix, then there is a basis for R n consisting ofeigenvectors for A.(c) det A t = det A. Hint. Are these defined?⎡6. Let A = ⎣ 2 1 6⎤ ⎡1 3 1⎦.Isv=⎣ 1 ⎤1⎦an eigenvector for A? Justify your answer.2 2 517. (a) The characteristic equation of⎡A = ⎣ 2 −4 1⎤0 3 0⎦1 −4 2is −(λ − 3) 2 (λ⎡− 1)=0. IsAdiagonalizable? Explain.(b) Is B = ⎣ 1 2 3⎤0 4 5⎦ diagonalizable? Explain.0 0 68. Let A be an n × n matrix with the property that the sum of all the entries ineach row is always the same number a. Without using determinants, show that thecommon sum a is an eigenvalue. Hint: What is the corresponding eigenvector?