Geometric interpretations of matrix inverses 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?
 A: Here is a visual answer for the $2\times 2$ case.  


*

*Plot the row (or column) vectors, $a_1, a_2$ of $A$ in $\mathcal{R}^2$ to visualize $A$. The area of the parallelogram they form is of course $\det(A)$.

*In the same space, plot the row (or column) vectors $a^1, a^2$ of $A^{-1}$, and the area of their parallelogram is then $\det(A^{-1}) = 1/ \det(A) $.

*The relationship between the two illustrates various properties of the matrix inverse.


An example is shown in the picture below, which comes from the matrix (in R notation)
A <- matrix(c(2, 1, 
              1, 2), nrow=2, byrow=TRUE)

In the R package matlib I recently added a vignette illustrating this with the following diagram for this matrix.

Thus, we can see:


*

*The shape of $A^{-1}$ is a $90^o$ rotation of the shape of $A$.

*$A^{-1}$ is small in the directions where $A$ is large

*The vector $a^2$ is at right angles to $a_1$ and $a^1$ is at right angles to $a_2$

*If we multiplied $A$ by a constant $k$ to make its determinant larger (by a factor of $k^2$), the inverse would have to be divided by the same factor to preserve $A A^{-1} = I$.
I wondered whether these properties depend on symmetry of $A$, so here is another example, for the matrix A <- matrix(c(2, 1, 1, 1), nrow=2), where $\det(A)=1$.

It would be interesting to extend this to other properties and to the $3 \times 3$ case, which I leave to others. 
A: 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.

