Cramer's rule: Geometric Interpretation I have a question concerning Cramer's rule:
Let $A$ be a matrix and $A \cdot \vec x = \vec b$ a lineare equation. $A_i$ is the matrix $A$ where the i'th column is replaced by $\vec b$


*

*if $det(A) \neq 0$, then we have a unique solution

*if $det(A)=0$ and at least one $det(A_i) \neq 0$, we have no solution

*if $det(A)=0$ and all $det(A_i)=0$ we have infinitely many solutions [false!]


I'm looking for a geometric interpretation of the rule. I know that $det(A)=$area of parallelepiped, but I'm not able draw a picture for Cramer's rule.
Anyone can help me here?
Thanks a lot in advance,
 A: One way to see Cramer's rule is that it simply makes use of a (very inefficient) way of calculating $A^{-1}$, specifically \begin{equation}A^{-1}=\frac{1}{\text{det}(A)}\text{Adj}(A),\end{equation} where Adj($A$) is known nowadays as the adjugate of $A$ (transpose of matrix of cofactors of $A$). If you substitute this equation into $X=A^{-1}B$ you basically get Cramer's rule, since for every component $x_i$ of $X$ you get the Laplace expansion by the ith column of $1/$det$(A)\cdot$det$(A_{C_i\leftrightarrow B})$ (you can check this...). 
So then to interpret this geometrically it depends on how far you want to go. If you just see it as a way of calculating $A^{-1}B$ then it simply means that a vector $X$ which is transformed by $A$ to give the vector $B$, if this transformation is invertible then you can calculate $X$ by applying the inverse transformation to $B$. If you now want to go further you have to try and explain geometrically why \begin{equation}A^{-1}=\frac{1}{\text{det}(A)}\text{Adj}(A).\end{equation} So in terms of a geometrical interpretation, we know $A$ is a transformation, and det($A$) is the area of the parallelogram formed by the component vectors in $A$. So what can be interesting is to explore the role of the adjugate and how it determines the formulae for det$(A)$...if you want to...I'll leave this for you - in 2D it's simple, since the cofactors of each entry is just another entry (with a sign change where needed).    
