If A and B are two matrices such that AB - A and BA - B- thenA- A 3- B 35- A 5- B 53B- A 4- B 43- A 12- B 12C- A - B is idempotentD- A 10- B 10 is nilpotent

The correct options are A (A^3 B^3)^5 = (A^5 B^5)^3 D (A^10 B^10) is nilpotentAB = A ⇒ ABA = A^2 ⇒ A^2 = A Similarly B^2 = B A and B are idempotent Also ( ...

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Byju's Answer

Standard XII

Mathematics

Nilpotent Matrix

If A and B ar...

Question

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

(A^{3} - B^{3})^{5} = (A^{5} - B^{5})^{3}

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(A^{4} - B^{4})^{3} = A^{12} - B^{12}

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(A - B)

is idempotent

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(A^{10} - B^{10})

is nilpotent

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Solution

The correct options are
A $(A^{3} - B^{3})^{5} = (A^{5} - B^{5})^{3}$
D $(A^{10} - B^{10})$ is nilpotent
$A B = A \Rightarrow A B A = A^{2} \Rightarrow A^{2} = A$
Similarly $B^{2} = B$
$A$ and $B$ are idempotent
Also $(A - B)^{2} = (A - B) (A - B)$
$= A^{2} - A B - B A + B^{2} = A - A - B + B = 0$
$(A - B)$ is nilpotent

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

The correct options are A (A3−B3)5=(A5−B5)3 D (A10−B10) is nilpotentAB=A⇒ABA=A2⇒A2=A Similarly B2=B A and B are idempotent Also (A−B)2=(A−B)(A−B) =A2−AB−BA+B2=A−A−B+B=0 (A−B) is nilpotent

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