More on Matrices - 3D Math Primer for Graphics and Game Development - page 171

• The inverse of a matrix product is equal to the product of the inverses

of the matrices, taken in reverse order:

−1 = B −1 A −1 .

( AB )

This extends to more than two matrices:

−1 = M n −1 M n−1 −1 M 2 −1 M 1 −1 .

( M 1 M 2 M n−1 M n )

• The determinant of the inverse is the reciprocal of the determinant of

the original matrix:

−1

M = 1/| M |.

6.2.3 Matrix Inverse—Geometric Interpretation
The inverse of a matrix is useful geometrically because it allows us to com-
pute the “reverse” or “opposite” of a transformation—a transformation
that “undoes” another transformation if they are performed in sequence.
So, if we take a vector, transform it by a matrix M , and then transform
it by the inverse M

−1 , then we will get the original vector back. We can
easily verify this algebraically:

−1 = v ( MM −1 ) = vI = v .

( vM ) M

6.3 Orthogonal Matrices

Previously we made reference to a special class of square matrices known
as orthogonal matrices. This section investigates orthogonal matrices a bit
more closely. As usual, we first introduce some pure math ( Section 6.3.1) ,
and then give some geometric interpretations ( Section 6.3.2) . Finally, we
discuss how to adjust an arbitrary matrix to make it orthogonal (Sec-
tion 6.3.3).

6.3.1 Orthogonal Matrices—Official Linear Algebra Rules
A square matrix M is orthogonal if and only if 1 the product of the matrix
and its transpose is the identity matrix:

1 The notation “P ⇔ Q” should be read “P if and only if Q” and denotes that the
statement P is true if and only if Q is also true. “If and only if” is sort of like an equals
sign for Boolean values. In other words, if either P or Q are true, then both must be
true, and if either P or Q are false, then both must be false. The ⇔ notation is also like
the standard “=” notation in that it is reflexive. This is a fancy way of saying that it
doesn't matter which is on the left and which is on the right; P ⇔ Q implies Q ⇔ P .