One small problem:

my @product = matrix_multiply(@matrix1, @matrix2);
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should be

my @product = @{ matrix_multiply(\@matrix1, \@matrix2) };
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The former is the same as:

my @product = matrix_multiply(
   $matrix1[0],
   $matrix1[1],
   $matrix1[2],
   $matrix2[0],
   $matrix2[1],
   $matrix2[2]
);
# @product has only one element.
# $product[0] is a ref to the resulting matrix
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