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xicor
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Homework Statement
[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 A4
Homework Equations
A = PDP-1,
AP = DP
A-I[tex]\lambda[/tex] = 0
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 v1=[0,-1,1] from [tex]\lambda[/tex]=2 , and {v2, v3} = {[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.