A question about a notation Let $A$ be a non-singular square matrix. Which of the following notations is correct? 
$${A^2}^{-1} \qquad \text{or} \qquad A^{-2}$$

I assume that the first one is interpreted as $(A^2)^{-1}$. Note that $A^{-2}$ is defined as $(A^{-1})^2$. Hence, if you want to interpret the two as meaning the same thing you have to think about it:
Exercise: Determine whether or not $(A^2)^{-1}=(A^{-1})^2$ for all non-singular square matrix $A$.
