is it idempotent?

[bhuvaneshnick]

if A and B are square matrices of the same order such that A*B=A and B*A=A then A and B are both _______ ?
This was question asked by my professor.Every one saying the answer as idempotent .But what i come up with is A*B=Band B*A=A


example


let


A= \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}

B= \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}


Both are idempotent as its clear from looking at it
By doing A*B i am getting B while doing B*A i am getting A


Could you help me discover What matrix A and B.THank you

 

[Ishuda]

How do you get to
A*B=B and B*A=A
from
A*B=A and B*A=A

A single example may help in discovering a proof of something but a single example never proves something for all cases.


From
A*B=B and B*A=A
it is moderately easy to get to both A and B are idempotent, for example
https://www.ma.utexas.edu/users/olenab/Fall-2011-341/341hwk2.pdf
or start playing around and, for example to prove A idempotent, multiply the first equation by A on the right and B on the left.