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Math 2270 Quiz 12 1. Find the eigenvalues of the matrix A= 6 2 3 7 . We must have 0 = det(A − λI) = det 6−λ 3 2 7−λ ⇒ λ2 − 13λ + = (6 − λ)(7 − λ) − 6 = 42 − 6λ − 7λ + λ2 − 6 = λ2 − 13λ + 36. 169 169 = −36 + ⇒ 4 4 2 25 13 13 5 = λ− ⇒ λ= ± = 9 or 4. 2 4 2 2 2. Find a basis for each eigenspace of the matrix in problem 1. From problem 1, we know that the eigenvalues are λ1 = 9 and λ2 = 4. To find the corresponding eigenvectors, we row reduce: 6−9 3 −3 3 x=r 1 −1 1 = ∼ ⇒ ⇒ ~v1 = , 2 7−9 2 −2 y=r 0 0 1 6−4 2 3 7−4 = 2 3 2 3 ∼ 1 0 3 2 0 Thus, {~v1 } is a basis for Eλ1 and {~v2 } is a basis for Eλ2 . ⇒ x = − 23 r y=r ⇒ ~v2 = −3 2 .