If $A$ is an $n \times n$ real matrix and
$$a_{ij}=\max(i,j)$$
for $i,j = 1,2,\dots,n$, calculate the determinant of $A$.
So, we know that
$$A=\begin{pmatrix}
1 & 2 & 3 & \dots & n\\
2 & 2 & 3 & \dots & n\\
3 & 3 & 3 & \dots & n\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
n& n & n & \dots & n
\end{pmatrix}$$
but what do I do after?
Best Answer
Let $d_n$ be the determinant of the $n\times n$ matrix
We can also write it as a recurrence
By expanding on the last row (or column) we observe that all but the minors of last two columns have linear dependent columns, so we have:
$d_n=-\frac{n^2}{n-1}d_{n-1}+nd_{n-1}=-\frac{n}{n-1}d_{n-1}$
Coupled with $d_1=1$ we get $d_n=(-1)^{n-1}n$