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

One can of course prove this directly by substituting and computing the entries of the $2 \times 2$ matrix on the left-hand side, but here's an outline for a solution that reduces the amount of actual computation one needs to do.
Hint


*

*Prove the claim for diagonal matrices.

*Prove the claim for matrices similar to diagonal matrices. (Recall that the trace and determinant of a matrix, which up to sign are the linear and constant terms of the polynomial in $A$, are invariant under similarity.)

*Use the fact that the set of diagonalizable matrices (those similar to diagonal matrices) is dense in the set of all $2 \times 2$ matrices.


Remark This is anyway a special case of a more general fact: That any matrix $A$ satisfies $p_A(A)$, where $p_A$ is the characteristic polynomial $p_A(t) := \det(t I - A)$ of $A$.
