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**Is there a faster way to do this matrix problem??**

## Homework Statement

Verify that [tex]\mathbf{A}[/tex] and [tex]\mathbf{\hat{A}} = \mathbf{P}^{-1}\mathbf{A}\mathbf{P}[/tex] have the same spectrum.

## Homework Equations

[tex]\mathbf{A} =

\left[

\begin{array}{ccc}

-22 & 20 & 10 \\

-4 & 20 & -8 \\

28 & -14 & 29 \\

\end{array} \right][/tex]

[tex]\mathbf{P} =

\left[

\begin{array}{ccc}

1 & 0 & 2 \\

0 & 2 & 4 \\

2 & 8 & 0 \\

\end{array} \right][/tex]

## The Attempt at a Solution

The problem is asking whether two similar matrices have the same set of eigenvalues.

Conceptually, I would first find the eigenvalues of A, by finding the characteristic equation, which will be a cubic equation resulting in 3 eigenvalues as the solutions. In this case, I used a computer to find [tex]\lambda = 36, 18, -27[/tex]. I then find the eigenvectors of A to verify that the matrix P represents the eigenvectors of A.

After that, I would compute the inverse of P using Gauss-Jordan elimination, then multiply the matrices out to find the similarity transform of A. Then, I would do the same method previously stated to find the eigenvalues of A.

My question is: Is there ANY nice, fast way to do all of this by hand? This seems like an extremely arduous process. The first time I found the eigenvectors of A, it resulted in filling up a page with text 4 times (then erasing) and finally getting the right answer on the 5th attempt, purely due to my error rate with the arithmetic.

I surely must not be properly understanding the concept if I am doing all this work to achieve the answers.