Show that a matrix $A=\pmatrix{a&b\\c&d}$ satisfies $A^2-(a+d)A+(ad-bc)I=O$ 
Let $A=
        \begin{bmatrix}
        a & b  \\
        c & d
        \end{bmatrix}
,a,b,c,d\in\mathbb{R}$ . Prove that every matrix $A$ satisfies the condition $$A^2-(a+d)A+(ad-bc)I=O .$$ Find $$        \begin{bmatrix}
        a & b  \\
        c & -a
        \end{bmatrix}^n .$$

For $a=1,b=2,c=3,d=4$ equality $A^2-(a+d)A+(ad-bc)I=O$ holds.
How to prove this equality for every $A=
        \begin{bmatrix}
        a & b  \\
        c & d
        \end{bmatrix}$?

Note that 
$$
A^2= \begin{bmatrix}
a^2+bc&ab+bd\\
ac+cd&cb+d^2
\end{bmatrix}
$$
and
$$
(tr A)A=(a+d)
 \begin{bmatrix}
a&b\\
c&d
\end{bmatrix}=
 \begin{bmatrix}
a^2+ad&ab+bd\\
ac+cd&ad+d^2
\end{bmatrix}
$$
so
$$
(tr A)A-A^2=
 \begin{bmatrix}
ad-bc&0\\
0&ad-bc
\end{bmatrix}
=
(det A) I
$$
Also note that 
$$
t^2-(tr A)t +det A=0
$$
is the characteristic equation of the $2\times 2$ matrix $A$ and, by Cayley-Hamilton theorem a matrix satisfies his characteristic equation.
