0

I need help with this problem:

For every $i,j \in \{1,2,\dots,n\}$ is $d_{i,j} = \min\{i,j\}$. Calculate determinant of a matrix $[d_{i,j}]_{n_\times n}$.

Is it right that all the elements of this squared matrix are $1$, because of that $d_{i,j}=min\{i,j\}$? Then, value of determinant is $0$.

Rodrigo de Azevedo
  • 1
user300045
  • 3,449
  • 1
    Why don't you just try a $2 \times 2$ example? What is $d_{2,2}$? – copper.hat May 20 '15 at 18:18
  • @ copper.hat But first I need to form a matrix. Is it right that all elements are $1$ by$d_{i,j}=min{i,j}$? – user300045 May 20 '15 at 18:20
  • I don't understand what you are asking. There is a formula for $d_{i,j}$ which gives the $i,j$ entry. – copper.hat May 20 '15 at 18:21
  • @ copper.hat I am asking what are the elements of a matrix? And what is that formula? – user300045 May 20 '15 at 18:23
  • The formula you gave above??? The $i,j$ entry is $\min(i,j)$. – copper.hat May 20 '15 at 18:24
  • Note that $d_{i,j}$ is not the minimum taken over all the possible values of $i$ and $j$ (which is $1$). The element $d_{i,j}$ is the $i$-th element of the $j$-th column (or the $j$-th element of the $i$-th row, as you prefer), and the minimum is taken only over those two indices. Thus, for example the $1$st element of every column is $1$, the $2$nd element of every column after the first is $2$, the $3$rd element of every column after the second is $3$, and so on (similarly for the rows). – A.P. May 20 '15 at 18:45
  • possible duplicate of Evaluation of a specific determinant. – Algebraic Pavel May 21 '15 at 09:25

1 Answers1

3

This matrix has determinant $1$, see here. The matrix looks as follows $$ \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 \cr 1 & 2 & 2 & \cdots & 2 \cr 1 & 2 & 3 &\cdots & 3 \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr 1 & 2 & 3 & \cdots & n \end{pmatrix} $$

Dietrich Burde
  • 130,978