Mathematical Prerequisites - Introduction to 3D Game Programming with DirectX 9.0 - page 15

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Inverses
In matrix math there is not an analog to division but there is a multipli-
cative inverse operation. The following list summarizes the important
information about inverses:
Only square matrices have inverses; therefore when we speak of
matrix inverses, we assume we are dealing with a square matrix.
The inverse of an n n matrix M is an n n matrix denoted as M -1 .
Not every square matrix has an inverse.
Multiplying a matrix with its inverse results in the identity matrix:
MM -1 = M -1 M = I . Note that matrix multiplication is commutative
when multiplying a matrix with its inverse.
Matrix inverses are useful for solving for other matrices in a matrix
equation. For example, consider the equation p = pR and suppose that
we know p and R and wish to solve for p . The first task is to find R -1
(assuming it exists). Once R -1 is known, we can solve for p , like so:

1 1

p R p RR

p R

1

pI

1

p R p

Techniques for finding inverses are beyond the scope of this topic, but
they are described in any linear algebra textbook. In the section titled
“Basic Transformations” we give the inverses for the particular matri-
ces that we work with. In the section titled “D3DX Matrices” we learn
about a D3DX function that finds the inverse of a matrix for us.
To conclude this section on inverses we present the following use-
ful property for the inverse of a product: ( AB ) -1 = B -1 A -1 . This
property assumes both A and B are invertible and that they are both
square matrices of the same dimension.

The Transpose of a Matrix
The transpose of a matrix is found by interchanging the rows and col-
umns of the matrix. Thus, the transpose of an m n matrix is an n m
matrix. We denote the transpose of a matrix M as M T .