Let $A=(a_{ij})$ be a $n \times n$ matrix where $a_{ij} = \max(i,j)$. Find the determinant of $A$. Let $A=(a_{ij})$ be a $n \times n$ matrix where $a_{ij} = \max(i,j)$. Find the determinant of $A$. How to find the solution of this kind of problem?
Hint: For $2\le k\le n$ substract the $k$-th row from the $k-1$-row.
Subtract the 1st row from the 2nd row we have the new 2nd row to be :

1 0 0 ... 0.
Expand the determinant on this 2nd row we have it equal to:

-1*/A/ where A is the new minor of the form:

2 3 4 5 ... n

3 3 4 5.... n

4 4 4 5 ... n

Repeat step 1 by subtracting 1st row of A from 2nd row of A and expand along the 2nd row we get: 

1*/B/ with B is the minor of A by deleting the 2nd row and 2nd column of A. Repeating this step on the way to the last step that we have: the final minor to be:

(n - 1)   n

n      n

this one has determinant = n(n - 1) - n*n  = -n. So the final answer is: -n if n = even, and n if n = odd.