Matrix inverses, determinants... questions

**1. If A is a square matrix satisfying ****$\displaystyle A^2 +10A = I$, where I is the identity matrix with the same dimensions as A, d****oes A have an inverse?**

I know the answer is yes, because I found the inverse, $\displaystyle 10I + A$, 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, & $\displaystyle \lambda$ is a number, what is the determinant of $\displaystyle \lambda A$?**

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

Thanks for any help