(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]

A=\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & -1\\ 0 & 0 & 2\end{array}

[/tex]

a) Find the eigenvalues and corresponding eigenvectors of matrix A.

b)Find the matrix P that diagonalizes A.

c)Find the diagonal matrix D suh that A = PDP^{-1}, and verify the equality.

d) Find the orthogonal matrix P that diagonalizes A.

e) Compute A^{4}

2. Relevant equations

A = PDP^{-1},

AP = DP

A-I[tex]\lambda[/tex] = 0

3. The attempt at a solution

First I started by finding the eigenvalues values where [tex]\lambda[/tex]=1 multipity two, 2. After this I tried finding the eigenvectors that form P and got v_{1}=[0,-1,1] from [tex]\lambda[/tex]=2 , and {v_{2}, v_{3}} = {[0, 1, 0], [0, 0, 1]}. From this I constructed the P matrix and got [tex]

P=\left[\begin{array}{ccc}0 & 0 & 0\\ -1 & 1 & 0\\ 1 & 0 & 1\end{array}

[/tex] and [tex]

D=\left[\begin{array}{ccc}2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}

[/tex] and this is where I get confused. The P matrix doesn't work in the form AP = PD and you can't find the inverse of P since the top row is all zeros. Once I figure this out, parts d and e should be straight-forward. Can someone point me to where I'm making a mistake here please. Thanks to everybody who helps.

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# Homework Help: Diagonalization help

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