Proving that if $AB=A$ and $BA=B$, then both matrices are idempotent

Question

Let $A, B$ be two matrices such that $AB=A$ and $BA=B$, how do I show that $A\cdot A=A$ and $B\cdot B=B$?

Steps I took:

1. Let $A= \left[\begin{array}{rr} a & b \\ c & d \\ \end{array}\right]$ and let $B= \left[\begin{array}{rr} w & x \\ y & z \\ \end{array}\right]$

2. $\left[\begin{array}{rr} a & b \\ c & d \\ \end{array}\right] \cdot \left[\begin{array}{rr} w & x \\ y & z \\ \end{array}\right] = \left[\begin{array}{rr} aw+by & ax+bz \\ cw+dy & cx+dz \\ \end{array}\right] $ (Which is also equal to A)

3. $\left[\begin{array}{rr} w & x \\ y & z \\ \end{array}\right] \cdot \left[\begin{array}{rr} a & b \\ c & d \\ \end{array}\right] = \left[\begin{array}{rr} wa+xc & wb+xd \\ ya+zc & yb+zd \\ \end{array}\right] $ (Which is also equal to B)

At this point I am stuck. I don't know how to proceed and I imagine that I started off on the wrong track to begin with. I'd like a hint to guide me in the right direction.

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My proof after consultation with answerers below:

Proof:

1) Since $AB=A$, we can say that: $(AB)A=AA$ which is equal to $A^2$

2) Then, (by associativity of matrix multiplication), we can say that $A(BA) = AB$ (since $BA=B$)

3) Then, $AB=A$ (since $AB=A$ was given)

4) Therefore, $AA$ is equal to $A$

5) Since $BA=B$, we can also say that: $(BA)B=BB$ which is equal to $B^2$

6) Then, (by associativity of matrix multiplication) we can say that $B(AB)=BA$ (since $AB=A$)

7) Then, $BA=B$

8) Therefore, $BB$ is equal to $B$

9) Thus, $AA$ is equal to $A$ and $BB$ is equal to $B$

Q.E.D.

Answer 1

If $AB = A$, and $BA = B$, then:

$A^2 = AA = (AB)A$ (since we can replace $A$ with $AB$ since they are equal)

$= A(BA)$ (by associativity of matrix multiplication)

$ = AB$ (since $BA = B$)

$= A$ (since $AB = A$).

Now you can try to use the same reasoning to prove $B^2 = B$.

This works for any two $n \times n$ matrices, $A,B$ that satisfy $AB = A$ and $BA = B$, not just $2 \times 2$ matrices.

Answer 2

Hint: Start with $AA = (AB)A$ (because of the first equality)

Answer 3

Notice that $AA = (AB)(AB) = A(BA)B = ABB = AB = A$.

Answer 4

$$A \cdot B = A \\ \implies (A \cdot B) \cdot A = A \cdot A \\ \implies A \cdot (B \cdot A) = A^2 \\ \implies A \cdot B = A^2 = A$$

Answer 5

Since $AB=A$ and $BA=B$, to show that $A$ is an idempotent we have to show that $A=A^2$ Therefore $A^2=(AB)^2 = A^2B^{ } = AB = A(BA) = AA = A^2 = A$. Proven