Question

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

(a) A
(b) I − A
(c) I + A
(d) 3A

Answer 1

$$\begin{gathered} {\left(A-I\right)}^3+{\left(A+I\right)}^3-7A \\ =A^3-I^3-3A^{2}I+3A{I}^2+A^3+I^3+3A^{2}I+3A{I}^2-7A \\ =2A^3+6A{I}^2-7A \\ =2A.A^2+6A-7A \\ =2A.I-A\left(\because A^2=I\right) \\ =2A-A \\ =A \end{gathered}$$

Hence, the correct option is (a).