det.md

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title: Determinants Made Easy author: Keith A. Lewis institution: KALX, LLC email: [email protected] classoption: fleqn abstract: Hermann Grassmann showed us how. ...

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.