More on Matrices - 3D Math Primer for Graphics and Game Development - page 170

Let's look at an example. In the previous section we calculated the
classical adjoint of a matrix M ; now let's calculate its inverse:

2
4 −4

3

−3 3

5

M =

0 2 −2 ;

1 4 −1

2 3 2

4 −1/4

3

6 9 0 −3/8 0

−1 = adj M
| M |

−24

4 5 5

M = −2 1 −8 = 1/12 −1/24 1/3 .

−2 13 −8 1/12 −13/24 1/3

Here the value of adj M comes from Equation (6.6), and | M | is from Equa-
tion (6.3).
There are other techniques that can be used to compute the inverse of a
matrix, such as Gaussian elimination. Many linear algebra textbooks assert
that such techniques are better suited for implementation on a computer
because they require fewer arithmetic operations, and this assertion is true
for larger matrices and matrices with a structure that may be exploited.
However, for arbitrary matrices of smaller order, such as the 2×2, 3×3, and
4×4 matrices encountered most often in geometric applications, the classical
adjoint method is generally the method of choice. The reason is that the
classical adjoint method provides for a branchless implementation, meaning
there are no if statements, or loops that cannot be unrolled statically. On
today's superscalar architectures and dedicated vector processors, this is a
big win.
We close this section with a quick list of several important properties
concerning matrix inverses.

• The inverse of the inverse of a matrix is the original matrix:

−1 ) −1 = M .

( M

(Of course, this assumes that M is nonsingular.)

• The identity matrix is its own inverse:

−1 = I .

I

Note that there are other matrices that are their own inverse. For
example, consider any reflection matrix, or a matrix that rotates 180 o
about any axis.

• The inverse of the transpose of a matrix is the transpose of the inverse

of the matrix:

( M T )

−1 = ( M −1 ) T