Interesting! This brings up the logical question: How would one perform A^7 on other HP's with built-in matrix support? Here are my answers:

HP-42S:

y^x is not supported for matrices. Don't use the x^2 function, because that will only square each matrix element. Instead, after loading the square matrix with the descriptor in the X-register:

ENTER, ENTER, ENTER, X, X, X, X, X, X

or,

ENTER, ENTER, X, ENTER, ENTER, X, X, X

HP-15C:

The result matrix cannot be stored to the same name (A, B, C, D, or E) as the name of either multiplicand matrix. With the square matrix loaded into "A":

RCL MATRIX A, ENTER, RESULT B, X, ENTER, RESULT C, X, RCL MATRIX B, RESULT D, X, RCL MATRIX A, RESULT B, X

A bit cumbersome, but there's enough memory to perform A^7 (or any n) for a 4x4 real-valued matrix A!

HP-41 with Math, Math/Stat, or Advantage module:

Suggestions? I have the eqipment, but haven't learned the added-on matrix functions for the 41.