# Q53 - Quiz 5.3 Using the information given below find a...

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Unformatted text preview: Quiz 5.3 Using the information given below, find a matrix P so that D = P- 1 AP is a diagonal matrix A = 6- 2- 7- 1 1 1 4- 2- 5 • Note that you do not have to compute P- 1 . • The characteristic polynomial for A is det( A- xI ) =- 2 + x + 2 x 2- x 3 . • x = 1 is a root of the characteristic polynomial. • The characteristic polynomial divided by ( x- 1) is- x 2 + x + 2 . SOLUTION: From the information we know that det( A- xI ) = ( x- 1)(- x 2 + x + 2). Further factoring tells us that 0 = det( A- xI ) = ( x- 1)(- x 2 + x +2) =- ( x- 1)( x 2- x- 2) =- ( x- 1)( x- 2)( x +1) ⇒ x =- 1 , 1 , 2 For each of these eigenvalues we must find an eigenvector which will provide a column for P . x = 1 A- I = 5- 2- 7- 1 1 4- 2- 6 R 1 ↔ R 2 = ⇒ - 1 1 5- 2- 7 4- 2- 6 R 1 ←- R 1 = ⇒ 1- 1 5- 2- 7 4- 2- 6 ( R 2 ,R 3) ← ( R 2- 5 R 1 ,R 3- 4 R 1) = ⇒ 1- 1- 2- 2- 2- 2 R 3 ←...
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Q53 - Quiz 5.3 Using the information given below find a...

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