Page 59 - Dynamics of Mechanical Systems

0593_C02_fm Page 40 Monday, May 6, 2002 1:46 PM

40 Dynamics of Mechanical Systems

Matrix products are distributive over addition and subtraction. That is,

(
AB C) = AB AC (2.10.6)
+
+
and

( BC A ) = BA CA (2.10.7)
+
+
Matrix products are also associative. That is, for conformable matrices A, B, and C, we
have:

( AB C ) = ( ABC (2.10.8)
A BC) =

Hence, the parentheses are unnecessary.
Next, it is readily seen that the transpose of a product is the product of the transposes
in reverse order. That is:

( AB) = B A T (2.10.9)
T
T
–1
Finally, if A is a nonsingular square matrix, the inverse of A, written as A , is the matrix
such that:

AA = A A = I (2.10.10)
−1
−1
–1
where I is the identity matrix. A may be determined as follows: Let M be the minor of
ij
the element a defined as the determinant of the matrix occurring when the ith row of A
ij
and the jth column of a are deleted. Let  be the adjoint of a defined as:
ij
ij
ˆ
A =− ( ) 1 ij + M (2.10.11)
ij ij
–1
Then A is the matrix with elements:
ij [ ] T
ˆ
A = A det A (2.10.12)
−1
T
where detA designates the determinant of A and [Â ] is the transpose of the matrix of
ij
adjoints.
If it happens that:
A = A T (2.10.13)
−1

then A is said to be orthogonal. In this case, the rows and columns of A may be considered
as components of mutually perpendicular unit vectors.