1. ## Matrices

Hi I'm not sure if this is the right forum to post this, but if it is not, I sincerely apologize.

I am really stuck at these two questions,

1.) Find all 2x2 matrices A for which A^2=A by letting A = $\begin{pmatrix} a & b \\c & d \end{pmatrix}$

This was the last part to this question, "The result "If ab=0 then a=0 or b=0 for real numbers does not have an equivalent result for matrices."
I don't know if this could be helpful but I really do not know how to do this question..

2.) Give one example which shows that if "A^2=O then A=O" is a false statement. (O is the null matrix!)

I got up to multiplying the matrices together and then equated it to 0..

Really appreciate your help!

2. ## Re: Matrices

if $A= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
then $A^2= \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} a^2+bc & ab+bd \\ ca+dc & cb+d^2 \end{pmatrix}$

so in 1) you have the four equation

$\\a= a^2+bc\\b= ab+bd\\c=ca+dc\\ d = cb+d^2$

So now you have to solve this system.

and as of for 2)

$\\0= a^2+bc\\0= ab+bd\\0=ca+dc\\ 0 = cb+d^2$

you must solve this system.
I gave an exemple (it is not unique) of a solution, but you should try to find it before checking.
Spoiler:

$\begin{pmatrix} \pi & -\pi \\ \pi & -\pi \end{pmatrix}$