Quiz11-Solution

# Quiz11-Solution - Name Math 331 Quiz 11 Thursday December 8...

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Name: .......................................................................................... Math 331 - Quiz 11 Thursday, December 8 1. Let A = 0 5 - 2 2 find an invertible matrix P and a rotation plus scaling matrix of the form C = a - b b a such that A = PCP - 1 . Solution: First we find the characteristic polynomial det ( A - λI ) = det - λ 5 - 2 2 - λ = ( - λ )(2 - λ ) + 10 = λ 2 - 2 λ + 10 To find the roots, we use the quadratic formula λ = 2 ± 4 - 40 2 = 2 ± - 36 2 = 2 ± 6 i 2 = 1 ± 3 i We choose one of the roots, for example 1 - 3 i . (Choosing 1 + 3 i would give a different, but also correct answer.) Then C = 1 - 3 3 1 To find P we must find an eigenvector. To do so we consider - (1 - 3 i ) 5 | 0 - 2 2 - (1 - 3 i ) | 0 = - 1 + 3 i 5 | 0 - 2 1 + 3 i | 0 Now we can choose either equation to find a complex eigenvector. I’ll use the first. (Choosing the second would give a different, but also correct answer.) Then we have ( - 1 + 3 i ) x 1 + 5 x 2 = 0 x 2 = - - 1 + 3 i 5 x 1 = - 1 - 3 i 5 x 1 So we can choose x 1 = 5 and then x 2 = 1 - 3 i . Then our complex eigenvector is x = 5 1 - 3

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