1. If A is a square matrix satisfying, where I is the identity matrix with the same dimensions as A, does A have an inverse?

I know the answer is yes, because I found the inverse, , basically by trial and error.

But is there a more elegant/systematic way to go about this in general?

2. If A is a square matrix with determinant d, & is a number, what is the determinant of ?

I know the answer, but I don't know why it is..

Thanks for any help