Page 51 - Aeronautical Engineer Data Book

38 Aeronautical Engineer’s Data Book
Singular matrix
A square matrix is singular if the determinant
of its coefficients is zero.

The inverse of a matrix
If A is a non-singular matrix of order (n 2 n)
–1
then its inverse is denoted by A such that AA –1
–1
= I = A A.
adj (A)
–1
A = 33 ∆ = det (A)
∆
A ij = cofactor of a ij
a 11 a 12 ... a 1n A 11 A 21 ... A n1
a 21 a 22 ... a 2n � � A 12 A 22 ... A n2 �
. . ... . –1 1 . . ... .
If A = A = 33
. . ... . ∆ . . ... .
� . . ... . . . ... .
a n1 a n2 ... a nn A 1n A 2n ... A nn

2.8.11 Solutions of simultaneous linear equations
The set of linear equations
a x + a x + ... + a x = b 1
12 2
1n n
11 1
a x + a x + ... + a x = b 2
2n n
21 1
22 2
� � � �
x + a x + ... + a x = b
a n1 1 n2 2 nn n n
where the as and bs are known, may be repre­
sented by the single matrix equation Ax = b,
where A is the (n 2 n) matrix of coefficients,
, and x and b are (n 2 1) column vectors.
a ij
The solution to this matrix equation, if A is
–1
non-singular, may be written as x = A b
which leads to a solution given by Cramer’s
rule:
= det D /det Ai = 1, 2, ..., n
x i i
where det D is the determinant obtained from
i
of the ith
det A by replacing the elements of a ki
(k = 1, 2, ..., n). Note
column by the elements b k
–1
that this rule is obtained by using A = (det A) –1
adj A and so again is of practical use only when
n ≤ 4.