📘 Question
If \(A\) and \(B\) are square matrices such that:
\[
AB = A \quad \text{and} \quad BA = B
\]
Prove that:
\[
A^2 = A \quad \text{and} \quad B^2 = B
\]
✏️ Step-by-Step Solution
Part 1: Prove \(A^2 = A\)
Given:
\[
AB = A
\]
Multiply both sides on the right by \(A\):
\[
ABA = A^2
\]
Using \(BA = B\):
\[
ABA = A(BA) = AB
\]
\[
= A
\]
So,
\[
A^2 = A
\]
Part 2: Prove \(B^2 = B\)
Given:
\[
BA = B
\]
Multiply both sides on the right by \(B\):
\[
BAB = B^2
\]
Using \(AB = A\):
\[
BAB = B(AB) = BA
\]
\[
= B
\]
So,
\[
B^2 = B
\]
✅ Final Result
\[
\boxed{A^2 = A \quad \text{and} \quad B^2 = B}
\]
💡 Key Concept
Matrices satisfying \(A^2 = A\) are called idempotent matrices. This problem shows both \(A\) and \(B\) are idempotent using given identities.