# Math Help - Proving Matrix

1. ## Proving Matrix

Sorry for this second question!

How do I prove that A^2 = A in Matrices given that AB=A and BA=A.
I tried to expand the LHS and then intend to get at AB so as to end up with RHS=A in the end but I'm stuckm
A^2 = A
AA = A
A(AB) = A
...

If I keep expandin, I'll keep getting more Bs and As..

Thank you!

2. ## Re: Proving Matrix

Originally Posted by Tutu
Sorry for this second question!
Don't be sorry it's a fun one!

I had start out saying that for A=0 it is pretty trivial that it work.

for A diff of 0 now, B can not be 0 since AB=A.

I'm pretty sure you can't go using only the matrice themself. There is no way you can prove that B or A as an inverse.
I think that you'll have to get your hand dirty and use the value in the matrice. Since we dont know the size you'll also have to work with a (NxN) matrices. You should first try with a 2x2 matrice. (I'm pretty sure it could work there if such a proof exist.)

Good luck!

3. ## Re: Proving Matrix

You are given AB=A and BA=A (for any matrix B I assume). Substitute B by A.

4. ## Re: Proving Matrix

Thank you both but I'm still stuck, can someone show me?

A^2
=AA
=A(AB)
A(A(BA))
= A^2(BA)
?

5. ## Re: Proving Matrix

Originally Posted by vincisonfire
(for any matrix B I assume).
Tutu is it for any matrix B? if so, A as to be the nul matrix. What is the full question?

6. ## Re: Proving Matrix

Originally Posted by Tutu
Sorry for this second question!

How do I prove that A^2 = A in Matrices given that AB=A and BA=A.
I tried to expand the LHS and then intend to get at AB so as to end up with RHS=A in the end but I'm stuckm
A^2 = A
AA = A
A(AB) = A
...

If I keep expandin, I'll keep getting more Bs and As..

Thank you!
Hi Tutu,

It is not true in general that if there are matrices A and B with AB = A and BA = A then A^2 = A.

For example, let

$A = \begin{pmatrix} 1 &0\\ 0 &2 \end{pmatrix} \quad B = \begin{pmatrix} 1 &0\\ 0 &1 \end{pmatrix}$

7. ## Re: Proving Matrix

Hi thank you all! Sorry for not including the whole question, didn't realize it was that important, my fault and apologies.

It is known that AB=A and BA=B where the matrices A and B are not necessarily invertible. Prove that A^2 = A (Note: From AB = A, you cannot deduce that B=I. Why?)

Adapted from IB Math HL Haese and Harris Publications