Linear Algebra: Find Matrix Determinant w/o Evaluating Directly

[NeedPhysHelp8]

Homework Statement

Find the determinant of the matrix using identities without evaluating the determinants directly:

a b b b
b a b b
b b a b
b b b a

The Attempt at a Solution

I tried getting it into a triangular matrix but halfway through it got too complicated and it has to be simpler than what I think it is. The matrix is symmetric but I don't know how that relates to the determinant.

 

[kikushiyo]

I don't know if you are supposed to know this way but let's try :
if a and b are real, a symmetric matrix is diagonalisable
You know that the eigenvalues are roots of any polynom that cancel your matrix. For a start, compute the square of you matrix and reexpress it in terms of the indentity matrix and your original one. This identity gies you a polynom that cancel your matrix. Its roots are eigenvalues, and their product is the determinant. Take care of the multiplicity of the eigenvalues, that's all !

You can also have guidance from basic aspects : you know trivial answers for special cases : a=b, a=0 or b=0 etc ... this should come then !

Last possibility (but also not very elegant) : trying linear combination of lines and you will find a nice factorization appearing...