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Here is my matrix. How do I find the determinant of this one? I'm really trying to solve it but I can't think of anything.

$$ \begin{pmatrix} 3 & 2& ...& 2\\ 2& 3& ...& 2\\ 2& 2& 3& ...\\ 2& 2& ...& 3 \end{pmatrix} $$

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asked many times, see math.stackexchange.com/questions/375711/… for example. Nevertheless, i suggest you calculate the $n$ by $n$ determinant for $n=1,2,3.$ Not hard –  Will Jagy Jun 17 at 18:30
    
That's the first thing I would do but I thought it wouldn't work for any number. –  user3601507 Jun 17 at 18:33
    
Please do $n=1,2,3$ anyway. Call it "hands-on experience." A few cases is not a proof, but it is more than you currently know, and the practice really will help. –  Will Jagy Jun 17 at 18:36
    
Alright I will try it now, thank you. Should I do it for X in the place of 3 and Y in the place of 2 for a more general approach? –  user3601507 Jun 17 at 18:36
    
I suggest $3,2$ first. –  Will Jagy Jun 17 at 18:38

1 Answer 1

up vote 0 down vote accepted

Hint:

Try to show that it is equal $5+2(n-2)$ using this. $$ \begin{vmatrix} 3 & 2& ...& 2\\ 2& 3& ...& 2\\ 2& 2& 3& ...\\ 2& 2& ...& 3 \end{vmatrix}=\begin{vmatrix} 1 & -1& 0&...& 0\\ 0& 1& -1& ...& 0\\ 0& 0& 1& ... &0\\ 2& 2& 2&...& 3 \end{vmatrix} $$

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