Determinants Matrices Question 167
Question: If A and B are two matrices such that $ AB=B $ and $ BA=A, $ then
Options:
A) $ {{(A^{5}-B^{5})}^{3}}=A-B $
B) $ {{(A^{5}-B^{5})}^{3}}=A^{3}-B^{3} $
C) $ A-B $ is idempotent
D) $ A-B $ is nilpotent
Show Answer
Answer:
Correct Answer: D
Solution:
- [d] Since AB=B and BA=A, so $ BAB=B^{2} $ Or $ (BA)B=B^{2} $ Or $ AB=B^{2} $ Or $ B=B^{2} $ Hence, B is idempotent and similarly A. $ {{(A-B)}^{2}}=A^{2}-AB-BA+B^{2} $
$ =A-B-A+B=0 $ Therefore, A-B is nilpotent.
BETA
