If A is an idempotent matrix and A - B -I- then which of the following is true- all of these-B is idempotent-AB -O-BA -O-

The correct option is A all of these A+B=I ⇒ A=I B Since A is idempotent, A^2=A ⇒(I B)^2=I B I^2 IB BI+B^2=I B ⇒ I 2B+B^2=I B ⇒ B^2=B i.e. B is idempotent ...

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Standard XII

Mathematics

Involutory Matrix

If A is an id...

Question

If A is an idempotent matrix and A + B =I, then which of the following is true?

B is idempotent

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$A B = O$

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$B A = O$

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all of these

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Solution

The correct option is D
all of these

$A + B = I \Rightarrow A = I - B S i n c e A i s i d e m p o t e n t, A^{2} = A \Rightarrow (I - B)^{2} = I - B I^{2} - I B - B I + B^{2} = I - B \Rightarrow I - 2 B + B^{2} = I - B \Rightarrow B^{2} = B i . e . B i s i d e m p o t e n t N o w, A^{2} = A (I - B) = A \Rightarrow A I - A B = A \Rightarrow A B = 0 a l s o, A^{2} = (I - B) A = A \Rightarrow I A - B A = A \Rightarrow B A = 0$

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If A is an idempotent matrix and A + B =I, then which of the following is true?

The correct option is D all of these A+B=I⇒A=I−BSince A is idempotent, A2=A⇒(I−B)2=I−BI2−IB−BI+B2=I−B⇒I−2B+B2=I−B⇒B2=B i.e. B is idempotentNow, A2=A(I−B)=A⇒AI−AB=A⇒AB=0also, A2=(I−B)A=A⇒IA−BA=A⇒BA=0