Matrix notation

Question

I was making matrix exercises when I ran into the matrix notation $X^{-2}$

Where $X$ is a $n \times n$ matrix. What is a matrix to the power $-2$?

Please provide me an answer because I couldn't find it anywhere else on the web.

Answer 1

In general for an $nxn$ matrix $A$, $A^k = A*A *...*A$ with the right hand side being multiplied $k$ times (which is well defined because of the associativity of matrixes).

In your case $A^{-2} = {(A^{-1})}^2 = A^{-1}*A^{-1}$, of course provided $A^{-1}$ exists.