📘 Question
If matrices \(A\) and \(B\) satisfy:
\[
AB = B \quad \text{and} \quad BA = A
\]
Find the value of:
\[
A^2 + B^2
\]
Options:
(a) \(2AB\)
(b) \(2BA\)
(c) \(A + B\)
(d) \(AB\)
✏️ Step-by-Step Solution
Step 1: Find \(A^2\)
From \(BA = A\), multiply on the left by \(A\):
\[
A(BA) = A^2
\]
\[
(AB)A = A^2
\]
Using \(AB = B\):
\[
BA = A
\Rightarrow A^2 = A
\]
Step 2: Find \(B^2\)
From \(AB = B\), multiply on the left by \(B\):
\[
B(AB) = B^2
\]
\[
(BA)B = B^2
\]
Using \(BA = A\):
\[
AB = B
\Rightarrow B^2 = B
\]
Step 3: Add results
\[
A^2 + B^2 = A + B
\]
✅ Final Answer
\[
\boxed{(c)\; A + B}
\]
💡 Key Concept
Using given identities cleverly reduces powers of matrices. Both \(A\) and \(B\) behave like idempotent matrices.