REVIEW OF MATRICES - Signals, Systems, and Transforms - page 742

C = A*B

AA -1 As an example, consider the product from (G.3) and (G.21):

11 1
11-1
21

-21 1
5
2

100
010
001

AA -1

1
2

= C S C - -1 S = C S . (G.27)

1
2

1
2

3 - 0

OTHER RELATIONSHIPS

Other important matrix relationships are now given.

Solution of Linear Algebraic Equations
Given the linear equations (G.1) expressed in vector-matrix format (G.4)

Ax = u ,

the solution is

x = A -1 u . (G.28)

M ATLAB performs this operation with the statement
x = inv(A)*u

For example, from (G.21), for the equations (G.1),

-21 1
5
2

3
1
6

1
1
1

x = A -1 u =

1
2

C - -1 SC S = C S . (G.29)

1
2

1
2

- 0

This solution is easily verified by substitution back into the original Equa-
tions (G.1).

Cramer's Rule
Given the n linear algebraic equations in vector-matrix form,

Ax = u ,

where A is n * n, x is n * 1, and u is n * 1, the i th component of x is given by

ƒ A i ƒ
ƒ A ƒ

x i = . (G.30)

In this equation, is the determinant of the matrix A , and is the determinant of
the matrix formed by replacing the i th column in A with the vector u . Equation (G.30)
is called Cramer's rule.

ƒ A ƒ ƒ A i ƒ