Determinant of an nxn Matrix

[ChaoticLlama]

I don't want the full solution for this question, a first step or two would be all. I'm not sure how to approach the problem (ie: how should I go about row-reducing to make calculating the determinant easier?):

What is the determinant of an nxn matrix with a main-diagonal of lambdas and r's everywhere else.

\(\displaystyle \L\left| {\matrix{
\lambda & r & r & r & \cdots & r \cr
r & \lambda & r & r & \cdots & r \cr
r & r & \lambda & r & \cdots & r \cr
\vdots & {} & \ddots & \ddots & \ddots & \vdots \cr
\vdots & {} & {} & \ddots & \ddots & r \cr
r & \cdots & \cdots & \cdots & r & \lambda \cr

} } \right|\)

For interest, the answer is \(\displaystyle \L\
[\lambda + (n - 1)r](\lambda - r)^{n - 1}\)

 

[pka]

You can try to do this by induction.
Start with a 2x2.

 

[ChaoticLlama]

thanks, i have the answer now.