Question

Choose the correct answer in the following question:
If A is square matrix such that $A^2=A,then(I+A{)}^3-7A$ is equal to
(a)A
(b)I - A
(c) I
(d)3A

Answer 1

Here, $A^2=A$
$\therefore (I+A{)}^3-7A=I^3+A^3+3I^{2}A+3A^{2}I-7A[\because (a+b{)}^3=a^3+b^3+3a{b}^2+3a^{2}b]=I+A^3+3A+3A^2-7A=I+A^2.A+3A+3A-7A[\because A^2=A]=I+A.A-A=I+A-A[\because A^2=A]$
$=I\therefore (I+A{)}^3-7a=I$. Hence, (c) is the correct option.