Prove the following using the properties of matrix

Question

Show that the matrix $$A=\begin{pmatrix} a&b\\c&d\end{pmatrix}$$ satisfies the equation $$A^2-(a+b)A+ (ad-bc) I =0$$

Please help me prove this. I have hard time doing it.

Answer 1

$$A^2=\begin{bmatrix} a^2+bc & ab+bd\\ ac+cd & ad+bd \end{bmatrix}$$ $$-(a+d)A=\begin{bmatrix} -a^2-ad & -ab-bd\\ -ac-cd & -ad-d^2\end{bmatrix}$$ $$(bc-ad)I=\begin{bmatrix} ad-bc & 0 \\ 0 & bc-ad \end{bmatrix}$$ Add these three results to get $$A^2-(a+d)A+(bc-ad)I=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

Note for OP: the middle co-efficiens has to be $(a+d)$ and not $(a+b)$ as written by you. This may be the reason why you could not get it.