If $A$ is any matrix and $n$ is an integer, what is $A^n$? If $A$ is any matrix and $n$ is an integer, what is $A^n$? What I know is $A_n$ is a matrix of order $n$. It seems that the superscript symbol (integer) is not common. What does it imply?
 A: The superscript symbol typically means the product of $A$ with itself $n$-many times. So $$A^3 = A AA$$ for instance, in the same way that $$5^3 = 5 \cdot 5 \cdot 5$$ If negative and $A$ is invertible, then it is the inverse being raised to $|n|$; for instance, if $A$ is invertible, then $A^{-4}$ means $$A^{-4} = \left( A^{-1} \right)\left( A^{-1} \right)\left( A^{-1} \right)\left( A^{-1} \right)$$
in the same way that
$$5^{-4} = \left( 5^{-1} \right)\left( 5^{-1} \right)\left( 5^{-1} \right)\left( 5^{-1} \right)$$
(though you might be more accustomed to just writing $1/5$ instead of $5^{-1}$).

To demonstrate with some explicit matrices, we have
$$\begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}^3
= \begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}\begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}\begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}$$
and
$$\begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}^{-4}
= \left( \begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}^{-1} \right) \left( \begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}^{-1} \right) \left( \begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}^{-1} \right) \left( \begin{bmatrix}
2 & 3 \\ 0 & 4 \end{bmatrix}^{-1} \right)$$
