GATE Chemical 2015 | Question: 12

Question

For the matrix $\begin{pmatrix} 4 & 3\\ 3& 4 \end{pmatrix}$, if $\begin{pmatrix} 1\\ 1 \end{pmatrix}$ is an eigenvector, the corresponding eigenvalue is __________ .

Answer 1

$we get $(4-\lambda)(4-\lambda)-9 \Rightarrow \lambda^2-8\lambda-9$

$\lambda^2-8\lambda-\lambda+7\Rightarrow \lambda(\lambda-7)-1(\lambda-7)$

$\lambda-7$ and $ \lambda-1 $are factors of the equation,

we need to find the eigen vectors for the following eigen values
and for 7 we get the given EV
4-7=-3   3

3       4-7=-3

to get zero vector we need vector(1,1) for eigen value 7.