For each $n$ let $A_{n}$ be the $n \times n$ matrix defined by

$$\left(A_{n}\right)_{i j}= \begin{cases}i & i \leqslant j \\ j & i>j\end{cases}$$

What is $\operatorname{det} A_{n} ?$ Justify your answer.

[It may be helpful to look at the cases $n=1,2,3$ before tackling the general case.]