Question

.Find the determinant of the matrix${\mathit{M}}_{\mathbf{n}}\mathbf{=}\left[\begin{array}{ccccc}1& 1& 1& ...& 1\\ 1& 2& 2& ...& 2\\ 1& 2& 3& ...& 3\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& 2& 3& ...& n\end{array}\right]$for arbitrary $\mathit{n}$. (The$\mathit{i}\mathit{j}$ thentry of ${\mathit{M}}_{\mathbf{n}}$is the minimum of i andj).

Short answer

Therefore, the determinant of the given matrix is given by,

$\det M_n=1,\forall n\in \mathbb{N}$

Step-by-step solution

Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Given

Given matrix,

$M_n=\left[\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& 2& 2& \cdots & 2\\ 1& 2& 3& \cdots & 3\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& 2& 3& \cdots & n\end{array}\right]$

To find determinant

If we multiply the$(n-1)$-th row and add it to the$n$-th row, the$n$-th row will become

$\left[\begin{array}{llll}0& \dots & 0& 1\end{array}\right]$ , but the determinant will stay the same.

Now, using Laplace expansion along the n -th row, we can see that. Since, we conclude that

$\det M_n=\det M_{n-1}$

Since$\det M_1=1$we conclude that$\det M_n=1,\forall n\in \mathbb{N}$