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 $A=
        \begin{bmatrix}
        a & b  \\
        c & d
        \end{bmatrix}$?

You can really just prove this by brute force, and its not that complicated.
$A=
        \begin{bmatrix}
        a & b  \\
        c & d
        \end{bmatrix}$
so 
$A^2=
        \begin{bmatrix}
        a^2-bc & ab-bd  \\
        ac-cd & bc-d^2
        \end{bmatrix}$
$(a+d)A=
        \begin{bmatrix}
        (a+d)a & (a+d)b  \\
        (a+d)c & (a+d)d
        \end{bmatrix}$
$(ad-bc)I=
        \begin{bmatrix}
        (ad-bc) & 0  \\
        0 & (ad-bc)
        \end{bmatrix}$
just add them together and you get 
$$A^2-(a+d)A+(ad-bc)I=O$$
