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.

Let matrix M be [a b]

[c d]

Then multiply the equation by M in two ways: one time on the left and one time on the right:

[M]^(G+1) = [0 1] * [a b] = [c d]

[0 0] [c d] [0 0]

[M]^(G+1) = [a b] * [0 1] = [0 a]

[c d] [0 0] [0 c]

The left sides are equal, therefore the right sides are equal. That gives a=d and c=0.

Then M is of the form [a b]

[0 a]

If a!=0 then powers of M will never have zeros on the main diagonal. If a=0 then M^2 is the zero matrix. Therefore there are no matrix M which satisfies the equation with G>=2.