📘 Question
If matrices \(A\) and \(B\) satisfy:
\[
AB = A \quad \text{and} \quad BA = B
\]
Find \(B^2\).
Options:
(a) \(B\)
(b) \(A\)
(c) \(I\)
(d) \(0\)
✏️ Step-by-Step Solution
Step 1: Use given relation
\[
BA = B
\]
Step 2: Multiply both sides by \(B\) (right side)
\[
BAB = B^2
\]
Step 3: Use \(AB = A\)
From \(AB = A\), substitute:
\[
BAB = B A B = B (AB) = B A
\]
\[
= B
\]
Step 4: Final result
\[
B^2 = B
\]
✅ Final Answer
\[
\boxed{(a)\; B}
\]
💡 Key Concept
Carefully use given matrix identities and substitute step-by-step. This problem shows that \(B\) behaves like an idempotent matrix where \(B^2 = B\).