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.

Determinants?

Been a long time!   Charlie slipped this one in while I was playing around with this in Excel on another computer.

Haven't got a solution but re the format, ⌈  ⌉ ⌊ and ⌋  might be worthy considerations should such a format be needed in the future (works in my editor, unsure about Perplexus).

Solution-wise, I begin with a matrix:
0 0    and arrive at    0  0       and          0 0
0 1    by squaring      0 -1     by cubing   0 1.

Since the latter matrix is a 90º rotation of what is sought I have a sense that our initial matrix is something like my initial one, not necessarily in that array, but also raised to yet a higher power.

Playing with this further, and raising the power to 4 just seems to create a kind of flip-flop mirroring but sometimes with diagonally values being signed differently.

At the moment this appears to beg a proof to prove such a G may not exist.

Edited on June 20, 2009, 12:15 pm

Posted by brianjn on 2009-06-20 11:46:49