Question

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

A. B² = B and A² = A 

B. B² ≠ B and A² = A 

C. A²≠ A, B² = B 

D. A² ≠ A, B² ≠ B

Answer 1

(A). B² = B and A² = A

AB = A ----- (i) 

BA = B ------(ii) 

From equation (ii),

B × (AB) = B 

B²A= B

From equation (ii), 

B²A = BA

B² = B 

From equation (i), 

A × (BA) = A 

A²B = A 

From equation (i),

A²B = ABA² = A 

Hence, 

A² = A & B² = B. 

Option (A) is the correct answer.