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I'm having issues finding out how to find all of these values. Is there a specific pattern? A formula of some sort?

If $A = \begin{bmatrix} x & y \\ z & a\end{bmatrix}$ then $A^2 = \begin{bmatrix} x & y \\ z & a\end{bmatrix}$

Kroki
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George Innit
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Sep 25 '22 at 17:13

1 Answers1

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If $A^2=A$, then $A$ is similar to one of the following matrices $$ \left[\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}\right],\left[\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right],\left[\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}\right]. $$ This is because every $x$ may be written as $x=(I-A)x+Ax$ and $$ A\{(I-A)x\}=0,\;\; A\{ Ax\} = Ax. $$

Disintegrating By Parts
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