Question

If $ A $ is a square matrix such that $ A^2 $ = $ A $ , then $ (I-A)^3+A $ is equal to

• A. $ A $
• B. $ I-A $
• C. $ I $
• D. $ 3A $

Answer 1

Answer: (C)

Given that, $A^{2}=A$
Then, $(I-A)^{3}+A$
$=\left(I\right)^{3}+\left(-A\right)^{3}+3\left(I\right)\left(-A\right)^{2}+3\left(-A\right)\left(I\right)^{2}+A$
$=I-\left(A\right)^{3}+3A^{2}+3\left(-A\right)\left(I\right)^{2}+A$
$=I-\left(A\right)^{3}+3A^{2}-3A+A$
$\because I^{3}=I, IA=A$
$=I-A\left(A\right)^{2}+3\left(A\right)^{2}-3A+A$
$=I-A\left(A\right)+3A-3A+A$ $\left(\because A^{2}=A\right)$
$=I-A^{2}+A$
$=\left(I-A\right)+A \left(\because A^{2}=A\right)$
$=I$