Can I apply elementary row operation then find eigen values of a matrix? Suppose if a matrix is given as
$$ \begin{bmatrix} 
4 & 6\\
2 & 9 
\end{bmatrix}$$
We have to find its eigenvalues and eigenvectors.
Can we first apply elementary row operation . Then find eigenvalues.
Is their any relation on the matrix if it is diagonalized or not.

First compute the eigenvalues as follows:
$$
\begin{align} 
\det(\lambda I-A)&=\det\left(
\begin{bmatrix}
\lambda & 0 \\
0 & \lambda
\end{bmatrix}
-
\begin{bmatrix}
4 & 6 \\
2 & 9
\end{bmatrix}
\right)
= \det \left(
\begin{bmatrix}
\lambda-4 & -6 \\
-2 & \lambda-9
\end{bmatrix}
\right) \\
&=
(\lambda-4)(\lambda-9)-(-2)(-6) \\
&=
\lambda^2-13\lambda + 24
\end{align}
$$
The eigenvalues are then
$$
\begin{align}
\lambda_{1,2} &= \frac{13 \pm \sqrt{13^2-4\times 24} }{2} \\
&= \frac{13\pm \sqrt{73}}{2}
\end{align}
$$
Now to compute the eigenvector for $\lambda_1$, we need to solve the following
$$
\begin{align}
\begin{bmatrix}
\lambda_1 - 4 & -6 \\
-2 & \lambda_1 -9
\end{bmatrix}
&=
\begin{bmatrix}
\frac{13 + \sqrt{73} }{2} - 4 & -6 \\
-2 & \frac{13 + \sqrt{73} }{2} -9
\end{bmatrix} \\
&= 
\begin{bmatrix}
\frac{5 + \sqrt{73} }{2}  & -6 \\
-2 & \frac{-5 +\sqrt{73} }{2} 
\end{bmatrix}
\end{align}
$$
Apply Gaussian elimination (i.e. $R_2=R_2+2\frac{2}{5+\sqrt{73}}R_1)$, we get
$$
\begin{bmatrix}
\frac{5 + \sqrt{73} }{2}  & -6 \\
0 & 0
\end{bmatrix}
$$
So, $x_2$ is a free variable and let $x_2=s$, we 
$$
\begin{align}
\frac{5 + \sqrt{73} }{2} x_1 = 6x_2 \implies x_1 = \frac{12}{5+\sqrt{73}}s
\end{align}
$$
Finally, 
$$
x=
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
= 
\begin{bmatrix}
\frac{12}{5+\sqrt{73}} s \\ s
\end{bmatrix}
= 
s
\begin{bmatrix}
\frac{12}{5+\sqrt{73}}  \\ 1
\end{bmatrix}
$$
The eigenvector for $\lambda_1$ is 
$$
v_1 =
\begin{bmatrix}
\frac{12}{5+\sqrt{73}}  \\ 1
\end{bmatrix}
$$
Now you can compute the second eigenvector in the same scenario. 
