I came across the following problem

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> Let matrix A be a $n\times n$ square matrix such that $a\_{ij}$ = max{ i , j }. Prove that det(A) = $(-1)^{n-1}n$
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I have read the post [If $a\_{ij}=\max(i,j)$, calculate the determinant of $A$](https://math.stackexchange.com/questions/2013663/if-a-ij-maxi-j-calculate-the-determinant-of-a) and understood all the answers adressing the same problem.  

I just wanted to see an alternative method **which does not use row subtractions**.
As, the determinant fromulla $n(-1)^{n-1}$ is reminiscent of $\frac{d}{dx}(x^{n})$ Is there some way we could use differentiation of determinant technique? How would it be if one goes by the formulla of determinant in terms of permutations?