Question

If A and B are two matrices such that AB = B and BA = A, then

• A. $(A^6-B^5{)}^3=A-B$
• B. $(A^5-B^5{)}^3=A^3-B^3$
• C. $A-Bisidempotent$
• D. $A-Bisnilpotent$

Answer 1

The correct option is D $A-Bisnilpotent$

Since AB = B and BA = A
$\therefore$ A and B both are idempotent
$(A-B{)}^2=A^2-AB-BA+B^2=A-B-A+B=0$
$\therefore$ A - B is nilpotent