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I have no problems solving a determinant problem and getting to the principle diagonal but the case is different in a prove that question where it asks me to give a specific result, I have no idea how to start solving, I've tried searching for tips but with no luck so I found this website and I'm hoping I get help :)

for example, this problem:

Without expanding, prove that:

\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \\ \end{vmatrix} = abc ( $\frac 1a$ + $\frac 1b$ + $\frac 1c$ + 1 )

I don't need a solution, I need tips on how to deal with these kinds of questions, any help would be appreciated :)

khaled014z
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  • It's hard to know what you mean by "without expanding", and surely you should consider expanding by minors as a way to get "a specific result". This is the $3\times 3$ case of a determinant that is computed in the general $n\times n$ case and in the $4\times 4$ case in the two answers on this previous Question. – hardmath Apr 07 '17 at 20:23

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