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

[Anony523]

Given a [FONT=MathJax_Math-italic]n[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math-italic]n[/FONT]n×n[/FONT] matrix whose [FONT=MathJax_Main]([FONT=MathJax_Math-italic]i[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]j[/FONT][FONT=MathJax_Main])[/FONT][/FONT]-th entry is i or j, whichever smaller, eg.
\begin{pmatrix}1 & 1 & 1 & 1\\
1 & 2 & 2 & 2 \\
1 & 2 & 3 & 3\\
1 & 2 & 3 & 4 \end{pmatrix}
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 [FONT=MathJax_Math-italic]A[FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT] mirror.

 

[stapel]

Given a [FONT=MathJax_Math-italic]n[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math-italic]n[/FONT]n×n[/FONT] matrix whose [FONT=MathJax_Main]([FONT=MathJax_Math-italic]i[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]j[/FONT][FONT=MathJax_Main])[/FONT][/FONT]-th entry is i or j, whichever smaller, eg.
\begin{pmatrix}1 & 1 & 1 & 1\\
1 & 2 & 2 & 2 \\
1 & 2 & 3 & 3\\
1 & 2 & 3 & 4 \end{pmatrix}
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 [FONT=MathJax_Math-italic]A[FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT] mirror.

Okay; you were given here how to apply induction, and received other suggestions here. What have you done? Where did this lead? Where are you stuck?

Please reply showing your thoughts and efforts so far. Thank you!

 

[pka]

Given a [FONT=MathJax_Math-italic]n[FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math-italic]n[/FONT]n×n[/FONT] matrix whose [FONT=MathJax_Main]([FONT=MathJax_Math-italic]i[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]j[/FONT][FONT=MathJax_Main])[/FONT][/FONT]-th entry is i or j, whichever smaller, eg.
\begin{pmatrix}1 & 1 & 1 & 1\\
1 & 2 & 2 & 2 \\
1 & 2 & 3 & 3\\
1 & 2 & 3 & 4 \end{pmatrix}
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 [FONT=MathJax_Math-italic]A[FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT] mirror.

Look at the evaluation of 'fourth' determinate using cofactors along the fourth row.
\[\left| {\left( {\begin{array}{*{20}{c}}
1&1&1&1 \\
1&2&2&2 \\
1&2&3&3 \\
1&2&3&4
\end{array}} \right)} \right| = 4\left| {\begin{array}{*{20}{c}}
1&1&1 \\
1&2&2 \\
1&2&3
\end{array}} \right| - 3\left| {\begin{array}{*{20}{c}}
1&1&1 \\
1&2&2 \\
1&2&3
\end{array}} \right| + 2\left| {\begin{array}{*{20}{c}}
1&1&1 \\
1&2&2 \\
1&3&3
\end{array}} \right| - \left| {\begin{array}{*{20}{c}}
1&1&1 \\
1&2&2 \\
1&3&3
\end{array}} \right|\]

What you want to do is argue that this pattern holds up in the induction.