solved If B is a matrix such that B^2 =B and A=I-B then which of the following is correct?

Question : If B is a matrix such that B^2 =B and A=I-B then which of the following is correct?

1. A^2=A
2. A^2=I
3. AB=0
4. BA=0

Solution :

From the Given matrix equation \[ B^2 =B\]

we have \[B^2-B=0\]

\[ B\left(B-I\right)=0 \]

and using matrix equation A=I-B

\[-BA =0 \]

\[BA=0\]

Correct option is (4).