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Unformatted text preview: We have Ax = x . (a) A 2 x = A ( Ax ) = A ( x ) = Ax = 2 x . (b) Ax = x multiply both sides by A-1 to get A-1 Ax = A-1 ( x ) or, equivalently, x = A-1 x . Hence, -1 x = A-1 x . (c) ( A + I ) x = Ax + Ix = x + x = ( + 1) x . 5. Problem Set 6.2, p. 298, # 4 A = S-1 S . Here is the eigenvalue matrix and S is the eigenvector matrix of A . We have A + 2 I = S S-1 + 2 SS-1 = S ( + 2 I ) S-1 . So the eigenvector matrix of A + 2 I is S and the eigenvalue matrix is + 2 I . 1...
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- Fall '11