Matrices $\begin{pmatrix} a &b \\ c &d \end{pmatrix}\begin{pmatrix} x\\y \end{pmatrix}=k\begin{pmatrix}x\\y \end{pmatrix}$ If $\begin{pmatrix}
a &b \\ 
c &d 
\end{pmatrix}\begin{pmatrix}
x\\y \end{pmatrix}=k\begin{pmatrix}x\\y \end{pmatrix}$, prove that $k$ satisfies the equation $k^2-(a+d)k+(ad-bc)=0$.
If the roots of this quadractic equation are $\alpha$ and $\beta$, find the value of $\alpha+\beta$ and $\alpha \beta$ in terms of $a,b,c$ and $d$. Hence, or otherwise, prove that $\begin{pmatrix}
a &b \\ 
c &d 
\end{pmatrix}\begin{pmatrix}
b &b \\ 
\alpha+a &\beta-a 
\end{pmatrix}=\begin{pmatrix}
b &b \\ 
\alpha+a &\beta-a 
\end{pmatrix}\begin{pmatrix}
\alpha &0 \\ 
 0&\beta 
\end{pmatrix}$
Can anyone give me some hints for solving this question? Thanks

$Ax=kx$
Roots of the given equations are the eigen-values of $A$.
So sum of roots = trace (A) = a+b
Product of roots = det (A)= ad-bc
