Question

what is the eigenvector and eigen value of the matrix
3 1
1 4

Answer 1

To find the eigenvalue(s) \[det\left[\begin{matrix}3-\lambda & 1 \\ 1 & 4-\lambda \end{matrix}\right]=0\]\[(3-\lambda)(4-\lambda) -1=0\]\[\lambda^{2}-7\lambda+11=0\]using the quadratic formula to solve for the roots we get\[\lambda={1\over 2}\left[7 \pm \sqrt{5}\right]\]Give me a second to figure out the eigenvectors

Answer 2

thank you..................but wht is the eigen vectors

Answer 3

The eigenvectors are \[v_{1}=\left[\begin{matrix}{1\over 2}(7+\sqrt{5})\\1\end{matrix}\right]\]and\[v_{2}=\left[\begin{matrix}{1\over 2}(7-\sqrt{5})\\1\end{matrix}\right]\]Basically, you are solving \[\left[\begin{matrix}3-\lambda & 1 \\ 1 & 4-\lambda\end{matrix}\right]\left[\begin{matrix}x_{1}\\x_{2}\end{matrix}\right]=0\] for each eigenvalue. The work is too long to post here but there you go