DETERMINANTS


Let a and b be two real (or complex) numbers. Consider the determinants of the nXn square matrices 

   D_n = | a b b ... b | 
         | b a b ... b |
         | b b a ... b |
         | . . . ... . |
         | b b b ... a |

and 
   G_n = | b b b ... b | 
         | b a b ... b |
         | b b a ... b |
         | . . . ... . |
         | b b b ... a |

In particular D_1 = a, D_2 = a2 - b2, and G_1 = b, G_2 = b(a-b).


Then the following identities hold 

  D_(n+1) = a D_n - b n G_n
  G_(n+1) = b D_n - b n G_n

From these get the closed form for the determinants:
D_n = (a-b)n + n b (a-b)n-1
G_n = b (a-b)n-1 (/center>

Marco Corvi - 2012