More on Matrices - 3D Math Primer for Graphics and Game Development - page 168

how we can use skew boxes to visualize coordinate space transformations
in Section 4.2.) By signed area, we mean that the area is negative if the
skew box is “flipped” relative to its original orientation.
In 3D, the determinant is the volume of the parallelepiped that has the
transformed basis vectors as three edges. It will be negative if the object is
reflected (“turned inside out”) as a result of the transformation.
The determinant is related to the change in size that results from trans-
forming by the matrix. The absolute value of the determinant is related
to the change in area (in 2D) or volume (in 3D) that will occur as a result
of transforming an object by the matrix, and the sign of the determinant
indicates whether any reflection or projection is contained in the matrix.
The determinant of the matrix can also be used to help classify the type
of transformation represented by a matrix. If the determinant of a matrix is
zero, then the matrix contains a projection. If the determinant of a matrix
is negative, then reflection is contained in the matrix. See Section 5.7 for
more about different classes of transformations.

6.2 Inverse of a Matrix

Another important operation that applies only to square matrices is the
inverse of a matrix. This section discusses the matrix inverse from a math-
ematical and geometric perspective.
The inverse of a square matrix M , denoted M

−1

is the matrix such

−1 on either side, the result is the identity

that when we multiply M by M

matrix. In other words,

−1 ) = M −1 M = I .

M ( M

Matrix inverse

Not all matrices have an inverse. An obvious example is a matrix with
a row or column filled with 0s—no matter what you multiply this matrix
by, the corresponding row or column in the result will also be full of 0s. If
a matrix has an inverse, it is said to be invertible or nonsingular. A matrix
that does not have an inverse is said to be noninvertible or singular. For any
invertible matrix M , the vector equality vM = 0 is true only when v = 0 .
Furthermore, the rows of an invertible matrix are linearly independent, as
are the columns. The rows (and columns) of a singular matrix are linearly
dependent.
The determinant of a singular matrix is zero and the determinant of a
nonsingular matrix is nonzero. Checking the magnitude of the determinant
is the most commonly used test for invertibility because it's the easiest and
quickest. In ordinary circumstances, this is OK, but please note that the
method can break down. An example is an extreme shear matrix with basis
vectors that form a very long, thin parallelepiped with unit volume. This