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## Homework Statement

Consider the matrix A =

| 7 16 8|

|-1 0 -1|

|-2 -10 -3|

Show that A is diagonalizable. Find an invertible matrix P and diagonal matrix D and use the obtained result in order to calculate A^2 and A^3

## Homework Equations

Determinant equation I suppose.

D is the diagonal matrix, like identity matrix but constructed from the eigenvalues. The order of the eigenvalues must match the columns in P precisely.

## The Attempt at a Solution

Alright well I've done a huge chunk of this problem already. I found the eigenvalues to be 2, 3, and -1. Then I find my eigenvectors, and I come up with 2 per eigenvalue.

For λ = 2 I get [-2 1 0] and [-1 0 1]

For λ = 3 I get [-3 1 0] and [-1 0 1]

For λ = 1 I get [1 1 0] and [0 1 1]

So now I have to construct P from these. This is where I'm confused. Which eigenvectors from which eigenvalues do I use? I've tried several combinations to make AP = PD and I just can't do it. Everything comes out wrong. I suppose there's just something I'm not understanding. Please help.