Matrix determinant proof problem

[hayze2728]

Homework Statement

I have to proove that the determinant of :

x a a . . . a
a x a . . . a
. . .
. . .
a . . . a
a . . . . . x

If you get the idea (it's (n x n) with x's along the diagonal and a's everywhere else)

That it is (x + (n-1)a)(x - a)^(n-1)


I really don't have a clue how to do this so any hints appreciated.

 

[EnumaElish]

Do you know how to compute the determinant?

 

[HallsofIvy]

Looks to me like induction on the size of the determinant would be best.

If n= 1, the determinant is just (x + (1-1)a)(x - a)^(1-1)= x.

Asume that formula is correct for a k by k determinant and evaluate the corresponding (k+1) by (k+1) determinant by expanding along the first row.