Q7ts1 - Quiz VII 101 1. The matrix A = 0 2 0 is symmetric....

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Quiz VII 1. The matrix A = 101 020 is symmetric. Find a basis for R 3 consisting of eigenvectors for A . Solution: Since A is symmetric, there will be an orthonormal basis consisting of eigenvectors of A , but you do not need to use that to solve the problem. First ±nd the eigenvalues det 1 λ 01 02 λ 0 10 1 λ =(1 λ ) 2 (2 λ ) (2 λ )=(2 λ )( 2 λ + λ 2 ) . So the eigenvalues are 2 and 0. Solving for the eigenspaces, ±v 1 = 1 0 1 is an eigenvector for the eigenvalue 0. For the eigenvalue 2, A 2 I = 10 1 000 1 , and the reduced row echelon form of that is just
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