M is a 2 x 2 matrix with each of the 4 elements being real. Can there exist an integer G ≥ 2, for which the following relationship is satisfied?

        [0    1]
M^G =|        |
        [0    0]

If the answer to the above question is "no", prove it. Otherwise, cite an appropriate example.

(In reply to Solution by Brian Smith)

Playing with your idea.  

M^n = [a^n  na^(n-1)b]

      [0    a^n      ]

If b=1/(2a^(n-1)) this becomes

M^n = [a^n 1  ]
      [0   a^n]

If a<1 this matrix tends to the [M]^G sought in the limit as n → ∞

(I know this isn't a solution.  I just thought it was interesting.]

Edited on June 22, 2009, 1:44 pm

Posted by Jer on 2009-06-22 13:37:20