# fast way to find the characteristic polynomial

I need to find the eigenvalues and eigenvectors of this matrix. $\left( \begin{array}{ccc} 7/34 & -11/34 & 4/17 & -1/17 \\ -11/34 & 27/34 & 1/17 & 4/17 \\ 4/17 & 1/17 & 31/34 & 5/34 \\ -1/17 & 4/17 & 5/34 & 3/34 \end{array} \right)$

Calculate the characteristic polynomial of this matrix is very long (there are no zeroes) so I would like to know which is the fastest way to find the eigenvalues and eigenvectors of this matrix.

The computer says the the characteristic polynomial is

$$t^2 (t-1)^2$$

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To make your calculations nicer you could multiply all entries by 34. This would have the effect of multiplying the eigenvalues by 34 too. –  Matt Gallagher Jun 24 '14 at 10:20

It's a hard to answer what is the fastest way, but if you see the definition of characteristic polynomial $\det(A-\lambda I)$, where $A$ is your given matrix, $I$ the identitity matrix, and $\lambda$ is the eigenvalues, what you called $t$. Then after Laplace expansion and simplification its easy to see, that the characteristic polynomial is $\lambda^4-2\lambda^3+\lambda^2$, what you can write into the form $\lambda^2(\lambda-1)^2$. After that the eigenvalues are $1, 1, 0, 0$.