Part 1: Prove $AB = A$ and $BA = B$ implies $A$ and $B$ are idempotent

Definitions:

• A matrix $A$ is idempotent if $A^2 = A$.

Given:

1. $AB = A$
2. $BA = B$

Proof for $A$ being idempotent:

1. Multiply $AB = A$ on the left by $A$: $A(AB) = A^2$
2. Substitute $AB = A$ into the equation: $A(A) = A^2 \quad \implies \quad A^2 = A$ Thus, $A$ is idempotent.

Proof for $B$ being idempotent:

1. Multiply $BA = B$ on the right by $B$: $(BA)B = B^2$
2. Substitute $BA = B$ into the equation: $B(B) = B^2 \quad \implies \quad B^2 = B$ Thus, $B$ is idempotent.

Part 2: Prove $A$ idempotent implies $A^n = A$ for all $n > 0$

Proof:

1. Start with $A$ idempotent: $A^2 = A$

2. Assume $A^k = A$ for some $k \geq 2$ (inductive hypothesis).

3. Show it holds for $k+1$: $A^{k+1} = A^k \cdot A$

4. By the inductive hypothesis, $A^k = A$: $A^{k+1} = A \cdot A = A$

5. Therefore, $A^n = A$ for all $n > 0$.

Final Results:

1. If $AB = A$ and $BA = B$, then both $A$ and $B$ are idempotent.
2. If $A$ is idempotent, then $A^n = A$ for all $n > 0$.

Follow-up Questions:

1. What are the geometric interpretations of idempotent matrices?
2. Can $AB = A$ and $BA = B$ occur if $A$ and $B$ are not square matrices?
3. How does the proof change if $A$ and $B$ are operators rather than matrices?
4. What are examples of idempotent matrices in practice?
5. Can the property $A^n = A$ be generalized to non-square matrices?

Tip: To verify idempotency, compute $A^2$ or $B^2$ directly and check if it equals the original matrix.