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$A=\begin{bmatrix}3 & -2 & 5\\ 1 & 0 & 7\\ 0 & 0 & 2\end{bmatrix}$, Find the eigenvalues of A.

I realized that if I swap columns I and II then I can make it an upper triangular matrix. Then the detrminant would be the product of the elements of the main diagonal. And then I would just need to find the roots of that.

However I know that swapping columns flips the sign of the determinant, but I don't know how that will effect finding the eigenvalues.

So I tried it anyways and got determinant of $(x+2)(x-1)(x-2)$ which has roots -2, 1, and 2. But I know that this is incorrect because the answers are supposed to be $\lambda=1,2,2$.

What did I do wrong?

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    To see that you can't swap columns, try finding the eigenvalues of $\pmatrix{1&0\cr0&1\cr}$ and $\pmatrix{0&1\cr1&0\cr}$, or of $\pmatrix{1&0\cr0&-1\cr}$ and $\pmatrix{0&1\cr-1&0\cr}$.2012-05-03

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