  It is widely known that the lower bound for the algorithmic complexity of
square matrix multiplication resorts to at least $n^2$ arithmetic operations.
The justification builds upon the following reasoning: given that there are $2
n^2$ numbers in the input matrices, any algorithm necessarily must operate on
each at least once. In this paper, we show that this is not necessarily the
case for certain instances of the problem, for instance matrices with natural
number entries. We present an algorithm performing a single multiplication and
$(n - 1)$ sums, therefore using n arithmetic operations. The ingenuity of the
approach relies on encoding the original $2n^2$ elements as two numbers of much
greater magnitude. Thus, though processing each of the inputs at least once, it
relies on a lower count of arithmetic operations. In the computational model
used to analyze this problem, such encoding operation is not available, thus it
is not clear this work affects the currently accepted complexity results for
matrix multiplication, but the new algorithm complexity (when taking into
account the encodings) is $3n^2 + 2n - 1$ operations. In addition, given the
exponential increase in multiplication operands magnitude, its practical usage
is constrained to certain instances of the problem. Nonetheless, this work
presents a novel mathematically inspired algorithm while pointing towards an
alternative research path, which opens the possibility of novel algorithms and
a taxonomy of matrix multiplications and associated complexities.
