Linear algebra - projection matrix - inverse matrix

Question

I am not sure how to prove this one:

Let $A$ be a projection matrix so that $A^2=A$ and $A$ is not equal to zero. Find the inverse matrix of $I+cA$.

Thanks.

Answer 1

If we search the inverse on the form $aI+bA$ we get

$$(aI+bA)(I+cA)=I\iff aI+(ac+b+bc)A=I\iff (a=1)\land(c+b+bc=0)\\\iff(a=1)\land (b=-\frac{c}{1+c}), c\ne-1$$