Question

If $B=B^2$ and $I-B=A$ then ______, ($A,B$ are square matrix of same order)

• A. $A^2=B$
• B. $A^2=I$
• C. $A^2=A$
• D. $A^2=-A$

Answer 1

The correct option is C

$A^2=A$

Explanation for the correct option

Given that $B=B^2$ and $I-B=A$

$$\begin{gathered} A=I-B \\ \Rightarrow A^2={\left(I-B\right)}^2 \\ \Rightarrow A^2=I^2-2BI+B^2 \\ \Rightarrow A^2=I-2B+B\left[\because I^2=I,BI=IB=B\mathrm{and}{\mathrm{B}}^2=\mathrm{B}\mathrm{given}\right] \\ \Rightarrow A^2=I-B \\ \Rightarrow A^2=A\left[\because A=I-B\mathrm{given}\right] \\ \therefore A^2=A \end{gathered}$$

Hence, as $A^2=A$, the correct option is option (C).