Question

If A is a square matrix such that A² = A, then write the value of 7A − (I + A)³, where I is the identity matrix.

Answer 1

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

Hence, the value of 7A − (I + A)³ is −I.