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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?

A

B is idempotent

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B

$A B = O$

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C

$B A = O$

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D

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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Involutory Matrix

Standard XII Mathematics

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