Question

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

• A. B is idempotent
• B. $AB=O$
• C. $BA=O$
• D. all of these

Answer 1

The correct option is D

all of these

$A+B=I\Rightarrow A=I-BSinceAisidempotent,A^2=A\Rightarrow (I-B{)}^2=I-B{I}^2-IB-BI+B^2=I-B\Rightarrow I-2B+B^2=I-B\Rightarrow B^2=Bi.e.BisidempotentNow,A^2=A(I-B)=A\Rightarrow AI-AB=A\Rightarrow AB=0also,A^2=(I-B)A=A\Rightarrow IA-BA=A\Rightarrow BA=0$