Determinant of matrix with Aij = min(i, j)

[nedf]

Given a n x n matrix whose (i,j)-th entry is i or j, whichever smaller, eg.
[1, 1, 1, 1]
[1, 2, 2, 2]
[1, 2, 3, 3]
[1, 2, 3, 4]
The determinant of any such matrix is 1.
How do I prove this? Tried induction but the assumption would only help me to compute the term for Ann mirror.

 

[Evgeny.Makarov]

From each line subtract the line immediately above it.

 

[nedf]

I will get this?
[1 1 1 1]
[0 1 1 1]
[0 0 1 1]
[0 0 0 1]
How do I prove that this matrix has same determinant as the original one?

 

[Evgeny.Makarov]

I will get this?
[1 1 1 1]
[0 1 1 1]
[0 0 1 1]
[0 0 0 1]

Yes.

How do I prove that this matrix has same determinant as the original one?

Subtracting another line is one of the elementary row operations and is known to preserve the determinant.