Can someone please provide an intuition behind cramer's rule?

Question

See question. I usually get concepts like this very quickly (no studying required), but this one looks like Chinese. Can someone please help me understand a brief intuition behind Cramer's rule for 2x2 and 3x3 matrices? TYVM

Answer 1

Suppose we want to solve $$x\begin{bmatrix}a_0\\a_1\end{bmatrix}+y\begin{bmatrix}b_0\\b_1\end{bmatrix}=\begin{bmatrix}c_0\\c_1\end{bmatrix}$$ for $x$ and $y$. Then we want a linear combination of vectors $\mathbf{a}$ and $\mathbf{b}$ such that they span a parallelogram with outer vertex at the point $c=(c_0,c_1)$ (draw a picture).

If we achieve this, the parallelograms spanned by

1) $y\mathbf{b}$ and $x\mathbf{a}$

2) $y\mathbf{b}$ and $\mathbf{c}$

will have the same area because they both have base $\mathbf{b}$ and height equal (consult picture to see why this is the case - for instance http://upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Cramer.jpg/400px-Cramer.jpg)

Because area is given by $2\times 2$ determinants, we get $$\det\begin{vmatrix}\mathbf{c}&y\mathbf{b}\end{vmatrix}=y\det\begin{vmatrix}\mathbf{c}&\mathbf{b}\end{vmatrix}=\det\begin{vmatrix}x\mathbf{a}&y\mathbf{b}\end{vmatrix}=xy\det\begin{vmatrix}\mathbf{a}&\mathbf{b}\end{vmatrix}$$ so that $$x=\frac{\det\begin{vmatrix}\mathbf{c}&\mathbf{b}\end{vmatrix}}{\det\begin{vmatrix}\mathbf{a}&\mathbf{b}\end{vmatrix}}$$ as required. This argument can be generalized to any dimension by considering the "volume" of the parallelopiped spanned by the vectors involved.