TITLE: Geometric interpretations of matrix inverses
QUESTION [19 upvotes]: Let $A$ be an invertible $n \times n$ matrix.  Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (this hyperplane does not intersect the origin).
Under this geometric interpretation, $A^{-1}$ has an interesting property: the normal vector to the hyperplane is given by the row sums of $A^{-1}$ (i.e. $A^{-1} \cdot 1$, where $1 = \langle 1, \dots, 1 \rangle^T$).
Within this geometric interpretation of $A$, what other interesting properties does $A^{-1}$ have?  Do the individual entries of $A^{-1}$ have geometric meaning?  How about the column sums?

REPLY [10 votes]: It turns out that an answer for the $3 \times 3$ case has similar properties and is also illuminating.

Start with a unit cube, representing the identity matrix. Show its
transformation by a matrix $A$ as the corresponding transformation of
the cube.
This also illustrates the determinant, det(A), as the volume of the
transformed cube, and the relationship between $A$ and $A^{-1}$.

In R, using the matlib and rgl package, the unit cube is specified as
library(rgl)
library(matlib)
# cube, with each face colored differently
colors <- rep(2:7, each=4)
c3d <- cube3d()
# make it a unit cube at the origin
c3d <- scale3d(translate3d(c3d, 1, 1, 1),
               .5, .5, .5)

A $3 \times 3$ matrix $A$ with $\det(A)=2$ is
A <- matrix(c( 1, 0, 1, 
               0, 2, 0,  
               1, 0, 2), nrow=3, ncol=3)

Extending the 2D idea from the answer above of drawing the images of $A$ and $A^{-1}$ together to 3D, we get the following, best viewed as an animated graphic. The faces of the parallelpiped representing $A^{1}$ are colored identically to those of $A$, so you can see the mapping from one to the other.