# 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?

-
Well, the column sum of $A^{-1}$ is the row sum of $(A^{-1})^T$, so since $A^T (A^{-1})^T = I$, there is a similar geometric interpretation. –  01000100 Feb 5 '13 at 8:21
Maybe this could be of some use: ofekshilon.com/2013/01/11/… –  PhD Oct 14 '13 at 18:57