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#MATL, 17 bytes

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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 n×ばつn:

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