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#MATL, 20 bytes

XJZyXyi:"!J21ドル!*s1$e

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###Explanation

This avoids the matrix multiplication doing it manually, by means of element-wise multiplication with broadcast. Specifically, to multiply matrices A and B, both of size s×ばつs:

1. Transpose A.
2. Permute the dimensions of B such that B is "turned" with a rotation axis along the first dimension, giving an s×ばつs 3D array, say C.
3. Multiply each element of A transposed times each element of C, with implicit broadcast. This means that A is automatically replicated s times along the third dimension, and C is replicated s times along the second, to make them both s×ばつs×ばつs before the actual element-wise multiplication
4. Sum along the first dimension to yield a ×ばつs×ばつs array.
5. Squeeze the leading singleton dimension out, to produce an s×ばつs result.

Commented code:

XJ % take matrix A. Copy to clipboard J
ZyXy % generate identity matrix of the same size
i: % take exponent n. Generate range [1 2 ... n] (possibly empty)
" % for each element in that range
 ! % transpose original A, or matrix with accumulated product
 J % paste A
 21ドル! % permute dimensions: rotation along first dimension axis
 * % element-wise multiplication with broadcast
 s1$e % squeeze out singleton dimension
 % end for. Display