Cramer's Rule, Volume and Transformations

Cramer's rule is simply a formula that solves a system of equations using nothing but determinants. It's named after Gabriel Cramer, the Swiss mathematician who published it in 1750. It's handy for small systems, for solving for just one unknown, and for proofs. The formula $x_i = \frac{\det A_i(\vec b)}{\det A}$ looks fancy, but it's one fraction: on the bottom, $\det A$ (the area of the parallelogram from $A$'s columns), and on top, $\det A_i(\vec b)$ (the same thing after you swap column $i$ out for $\vec b$). One determinant divided by another. That's all.

Why it works. Think about the area picture. Write $\vec b = x_1\vec a_1 + x_2\vec a_2$ and replace $\vec a_1$ with $\vec b$. The $x_2\vec a_2$ part of $\vec b$ points along the other edge, so it only shears the parallelogram, and shearing doesn't change area (that's row replacement from last lesson). What's left is $x_1\vec a_1$, which scales the area by exactly $x_1$. So the ratio of the two areas is $x_1$. There's an animation further down that shows this happening.

Cramer's rule is elegant for $2\times2$ and $3\times3$ systems and for theory, but slow for big systems, where row reduction wins.

Areas and volumes. Put the edge vectors in the columns, and the absolute value of the determinant is the area (2D) or volume (3D). A triangle is half a parallelogram.

Transformations scale volumes. If $T(\vec x) = A\vec x$, then any region's area or volume gets multiplied by $|\det A|$. This is exactly the Jacobian factor in multivariable change of variables.

A common mistake. It's tempting to replace a row of $A$ with $\vec b$, since $\vec b$ is written next to the rows of the system. It's a column: column $i$ holds the coefficients of $x_i$, and that's the column $\vec b$ takes over.

Computational geometry uses determinant signs constantly: to test which side of a line a point is on, whether a triangle is clockwise, and to compute polygon areas.