Matrix solutions to $A^k + B^k = C^k$
Does there exists any triple $(A,B,C)$ of $n\times n$ ($n\geq3$) matrices with integer entries which satisfies $A^k+B^k=C^k$ for at least one $k\geq3$? Here $n,k$ are given to us, that is, we can choose them according to our convenience.
There are basically trivial examples, for example with n=2: $A = e_{11}$, $B = e_{22}$, $C = I$ (where $e_{11}$ has a 1 in the (1,1) spot, zeroes elsewhere). What does this have to do with Sage? And how do you characterize "trivial" so as to exclude such examples?
An even more trivial example: all of the matrices could be zero, or $A=0$ and $B=C$.
actually n is strictly greater than 2 and none of A,B,C are non zero and non nilpotent matrix
A = diag(1,0,0),B = diag(0,1,0)andC = diag(1,1,0)works for all k.Repeating John question: what is the link with the Sage software?