Cramers Rule. The why and how.

Question

Can someone explain how Cramer's rule works. I understand the mechanics of it, and it's fairly straightforward to show algebraically that it's equivalent to GJ and substitution, but what's happening under the hood? I'm guessing it has to do with properties of determinants but...

Answer 1

Let $Ax=b$ be a system of $n$ linear equations in $n$ unknowns.
Let $a_j$ be the $j$-th column of $A$. Then $b = x_1 a_1 + x_2 a_2 + \cdots x_n a_n$.

Write $\det(c_1,c_2,\dots,c_n)$ for the determinant of a $n\times n$ matrix with columns $c_1,c_2,\dots,c_n$.

The key property of $\det$ is that is an alternating multilinear function of $c_1,c_2,\dots,c_n$.
This means that you can expand sums and take scalars out, and that $\det$ is zero if two columns are equal.

Now just compute:

$\ \det(a_1, \dots, b, \dots, a_n)= \det(a_1, \dots, x_1 a_1 + \cdots +x_n a_n, \dots, a_n)= $
$ \quad x_1\det(a_1, \dots, a_1, \dots, a_n) + \cdots $
$ \quad x_j\det(a_1, \dots, a_j, \dots, a_n) + \cdots $
$ \quad x_n\det(a_1, \dots, a_n, \dots, a_n) $
$ \quad = x_j \det(A) $

This is Cramer's rule.