Question

If A is a square matrix such that A² = A, then (I + A)³ − 7A is equal to
(a) A
(b) I − A
(c) I
(d) 3A

Answer 1

(c) I

$$\begin{gathered} \mathrm{Here}, \\ {A}^2=A...\left(1\right) \\ {A}^3=A^{2}A \\ =A^2\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right] \\ =A \\ \therefore A^3=A...\left(2\right) \\ \mathrm{We}\mathrm{know}\mathrm{that}{\left(I+A\right)}^3=I^3+3{\left(I\right)}^{2}A+3\left(I\right)A^2+A^3 \\ \Rightarrow {\left(I+A\right)}^3=I+3A+3A+A\left[\mathrm{From}\mathrm{eqs}.\left(1\right)\mathrm{and}\left(2\right)\right] \\ \Rightarrow {\left(I+A\right)}^3=I+7A \\ \Rightarrow {\left(I+A\right)}^3-7A=I \end{gathered}$$