Determinants Made Easy

What is the determinant of a square matrix A = [a_{ij}]? Let P_i\in E be points in Euclidean space E for i in a finite index set I. In 1844 Hermann Grassmann published Die Lineale Ausdehnungslehre that posited P_i P_j = -P_j P_i, for i,j\in I. If i = j then P_i P_i = -P_i P_i so 2P_i P_i = 0 and P_i^2 = 0.

If a is a number and P is a point then aP is a point with weight a. He assumed {aP + bP = (a + b)P} and {(aP)(bQ) = (ab)(PQ)} for numbers a,b and points P,Q.

That is all you need to define determinants. If {P_j = \sum_i a_{ij}P_i} then {\prod_i\sum_j a_{ij} P_i = (\det A)\prod_i P_i}.

\begin{aligned} (a_{11}P_1 + a_{12}P_2)(a_{21}P_1 + a_{22}P_2) &= a_{11}P_1 a_{21}P_1 + a_{11}P_1 a_{22} P_2 + a_{12}P_2 a_{21}P_1 + a_{21}P_2 a_{22} P_2\\ &= a_{11} a_{21}P_1 P_1 + a_{11}a_{22} P_1 P_2 + a_{12} a_{21}P_2 P_1 + a_{21} a_{22} P_2 P_2\\ &= a_{11}a_{22} P_1 P_2 + a_{12} a_{21}P_2 P_1\\ &= a_{11}a_{22} P_1 P_2 - a_{12} a_{21}P_1 P_2\\ &= (a_{11}a_{22} - a_{12} a_{21})P_1 P_2\\ \end{aligned} so \det[a_{ij}] = a_{11}a_{22} - a_{12}a_{21} in 2 dimensions.

The same computational technique holds in any number of dimensions for computing determinants. Back in Hermann’s day people were still grappling with how to extend Euclidiean geometry into higher dimensions. He was way ahead of his time.