Linear Algebra | Part IB, 2004

For each nn let AnA_{n} be the n×nn \times n matrix defined by

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

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

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

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