|(under the auspices of Prof. Robert L. Wilson)|
Suppose that we have a real-valued system of n linear equations with n unknowns:
Instinctively, we recast our system in terms of a single matrix equation of the form Ax = b:
Now, in view of the machinery of linear algebra, whether are we assured a unique solution is equivalent to asking whether A is invertible. One particularly concise characterization of invertibility for real, square matrices involves the notion of determinant, stating that the invertible matrices are those whose determinant is nonzero.
Recall that for the space of n×n matrices over the reals, the determinant function assigns to each matrix a real number by, very loosely speaking, "summing over all permutations" of the entries. More formally, we have the following definition:
That each product is listed with indices in ascending order is done solely in the interest of aesthetics; indeed commutativity of real multiplication assures us that the ordering of the factors is immaterial. Absent from our definition are parentheses since real multiplication is also associative, thus permitting us to group factors in whatever way we deem most convenient.
It is upon the conditions of associativity and commutativity that the determinant so heavily relies. How might we amend its definition to accommodate structures bereft of these algebraic amenities? In particular, we will address this problem for the non-associative, non-commutative Octonions, an 8-dimensional algebra over the reals.