You can do it with iterative calculation

• check the box: file - options - formulas - "enable iterative calculations" (from the version Excel 2007)
• set "maximum iterations" value to a number big enough (this is the maximum power you can raise the matrix)

• Formula in helper cell:
  =IF(ISNUMBER(G2),IF(G2=E2,TEXT(G2,"0"),G2+1),IF(VALUE(G2)=E2,G2,1))
• Formula in result matrix:
  =IF(ISNUMBER(G2),IF(G2=1,A2:C4,MMULT(A2:C4,H2:J4)),H2:J4)
  (this is an array formula, need to finish with CTRL+SHIFT+ENTER)

Notes

• The formula is recalculated ONLY if you CHANGE POWER, it won't be recalculated after changing only the input matrix, neither at recalculation of all formulas (could be done but requires more complex formula and more helper cells, let me know if you're interested in it).
• Also, after entering both formulas, output matrix will show only 0 values, to get it calculated you need to change the power, from that the sheet will be "initialized", it will work without problems
• error checking is not implemented in the formula yet:
  • if power > maximum number of iterations, than formula will run only the times of iteration set
  • similarly, for non-integer or negative numbers formula won't stop, but run until max number of iterations
  • after maximum number of iterations reached formula will "overflow", counts forward without starting from 0, to fix that, set a power which will be reached in the next iteration cycle (e.g. if helper = 500, max number of iteration = 100, then set a 500 < power < 600, so formula will correctly stop there, and it'll work correctly for next setting of power).
• majority of the issues described above can be fixed within the formula if necessary.

update

Answer to this question describe with more details how the formula works.

Confirmed: Also happens in MS Word 2010, 2013, 2016 all 64 bit Office*1

*1 I suspect this applies to most of, if not all, the 64 bit office applications

This seems to be an issue with the 64 bit version of MS Office. In the 32 bit version the caret ^ is used for the power operator, however in the 64 bit operator it is also used for variables of type LongLong (learn.microsoft.com/.../vba/.../longlong-data-type).

In 64 bit VBA, when using Debug.Print*2 the IDE auto corrects 2^2 to 2^; 2 but 2 ^2 is auto-corrected to 2 ^ 2; making the code compile-able.

By way of a solution, this is just one of those annoying things that users of 64bit VBA 'just need to know'.

Edit: Microsoft confirmed issue https://support.microsoft.com/en-gb/help/2454019/the-64-bit-office-2010-vba-compiler-shows-an-error-when-it-encounters

*2 I've also come to learn that the VBA behavior is different when using debug.Print than assigning a variable (thanks to @ErikvonAsmuth)