Question

If AB = A and BA = B, where A and B are square matrices,  then

Options

• B² = B and A² = A

• B² ≠ B and A² = A

• A² ≠ A , B^{2 =}B

• A² ≠ A , B² ≠ B

Answer 1

B² = B and A² = A 

\[Here, \] 

\[AB = A . . . \left( 1 \right) \] 
\[BA = B . . . \left( 2 \right)\] 
\[ \Rightarrow ABA = AA \left[ \text{Multiplying both sides by }A \right] \] 
\[BAB = BB \left[ \text{Multiplying both sides by }A \right] \] 

\[ \Rightarrow AB = A^2 \left[ \text{From eq} . \left( 2 \right) \right] \] 
\[BA = B^2 \left[ \text{From eq }. \left( 1 \right) \right]\] 
\[ \Rightarrow A = A^2 \left[\text{ From eq} . \left( 1 \right) \right] \] 
\[B = B^2 \left[ \text{From eq} . \left( 2 \right) \right]\]