$A$ is a square matrix with $A^2 = A$, and $(I + A)^3 = I + kA$. What is $k$?

Explanation

1. 01

   $I$ commutes with $A$, so $(I + A)^3 = I + 3A + 3A^2 + A^3$.

2. 02

   Every power of $A$ is $A$, so this is $I + 3A + 3A + A = I + 7A$, and $k = 7$.