Matrix solutions to $A^k + B^k = C^k$

asked 2021-04-22 20:02:36 +0100

anonymous user


updated 2021-04-23 11:17:39 +0100

slelievre gravatar image

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.

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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?

John Palmieri gravatar imageJohn Palmieri ( 2021-04-22 21:06:15 +0100 )edit

An even more trivial example: all of the matrices could be zero, or $A=0$ and $B=C$.

John Palmieri gravatar imageJohn Palmieri ( 2021-04-22 21:07:39 +0100 )edit

actually n is strictly greater than 2 and none of A,B,C are non zero and non nilpotent matrix

rewi gravatar imagerewi ( 2021-04-22 21:10:08 +0100 )edit

A = diag(1,0,0), B = diag(0,1,0) and C = diag(1,1,0) works for all k.

vdelecroix gravatar imagevdelecroix ( 2021-04-22 23:07:30 +0100 )edit

Repeating John question: what is the link with the Sage software?

vdelecroix gravatar imagevdelecroix ( 2021-04-22 23:08:18 +0100 )edit