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Q7rs1 - Quiz VII 201 1 The matrix A = 0 1 0 is symmetric...

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Quiz VII 1. The matrix A = 2 0 1 0 1 0 1 0 2 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 find the eigenvalues det 2 λ 0 1 0 1 λ 0 1 0 2 λ = (2 λ ) 2 (1 λ ) (1 λ ) = (1 λ )(3 4 λ + λ 2 ) . So the eigenvalues are 3 and 1. Solving for the eigenspaces, v 1 = 1 0 1 is an eigenvector for the eigenvalue 3. For the eigenvalue 1, A I = 1 0 1 0 0 0 1 0 1 , and the reduced row echelon form of that is just
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