If A is idempotent matrix and A + B = I then B is

Since A is Idempotent matrix then ${\mathrm{A}}^2=\mathrm{A}$

$$\begin{gathered} \therefore \mathrm{A}+\mathrm{B}=\mathrm{I} \Rightarrow \mathrm{B}=\mathrm{I}-\mathrm{A} \\ {\mathrm{B}}^2=(\mathrm{I}-\mathrm{A}) (\mathrm{I}-\mathrm{A}) \\ =\mathrm{I}-\mathrm{A}-\mathrm{A}+{\mathrm{A}}^2 \\ =\mathrm{I}-2\mathrm{A}+\mathrm{A} \\ =\mathrm{I}-\mathrm{A} \\ {\mathrm{B}}^2= \mathrm{B} \end{gathered}$$