# How to find eigenvector of given matrix?

Is Lambda=4 is eigenvalue of

$$\left( \begin{matrix} 3 & 0 & -1 \\ 2 & 3 & 1 \\ -3 & 4 & 5 \\ \end{matrix} \right)$$ If so find one corresponding eigenvector.

I know how to determine if the given lambda value is eigenvalue. I just solve $(A-\lambda I)x=0$. But I'm not sure what approach to use in finding eigenvector of that matrix.

Thanks

• I don't know a best way than an other to find eigenvector... – Hexacoordinate-C Mar 14 '15 at 19:20
• Set the determinant of the relevant matrix equal to zero? – Mark Bennet Mar 14 '15 at 23:00

## 1 Answer

when i row reduce the matrix $A - 4I$ i get $$\pmatrix{1&0&1\\0&1&1\\0&0&0}$$ therefore an eigenvector corresponding to the eigenvalue $4$ is $$\pmatrix{1\\1\\-1}.$$

• Thanks! But why is it -x3 as a common?Why not just x3? – user3273345 Mar 14 '15 at 19:32
• @user3273345, any nonzero multiple of an eigenvector is also an eigenvector. i just picked $x_3 = -1$ so that we can have $x_1 = x_2 = 1.$ had i picked $x_3 = 1,$ then $x_1 = x_2 = -1$ – abel Mar 14 '15 at 19:35