(a) Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible. What is the inverse?
(b) Show that if $p(x)$ is a polynomial with a nonzero constant term, and if $A$ is a square matrix for which $p(A) = 0$, then $A$ is invertible.
What am i supposed to do here? plug a square matrix with a b c d in the problem?.. but then what? and i dont have a clue how to do the second one either…
Best Answer
You could use that approach, but it sounds pretty miserable. Rather, consider the fact that
$$I = -A^2 - 2A = A(-A - 2I)$$
For the second part, something essentially the same will work: Move the constant term to the other side and factor out an $A$.