5.2 Properties of Determinants
In this section we look at the algebraic properties of determinants that are the key to
computing and applying them.
Property 1 Determinants of Triangular Matrics. As we saw in the previous section
the definition of a n n in
5 Determinants
The previous chapter on inverses of matrices gave us a formula for how the solution of a
system of linear equations depends on the right hand side. This chapter gives us a
formula for how the solution depends on the coefficients. The formul
5.3 Applications of Determinants
In this section we look at applications of determinants to solving equations and to areas
and volumes.
5.3.1 A Characterization of Invertible Matrices. One application of determinants is
that it provides a simple character
4.2 Algebraic Properties of Matrix Inverses
The operation of taking the inverse of a matrix has several algebraic properties that are
similar to taking the inverse of a number. However, with some of them one has to be
careful of the order that we multiply
3.2 Fitting Equations to Data
A common application of linear equations is to fit equations to data. We shall illustrate
this technique with an example.
Example 1. Find a quadratic equation
(1)
y = ax2 + bx + c
y
7
6
such that
4
y = 6 when x = 1
(2)
5
3
y
3.5
Uniqueness of Solutions to Linear Equations
Linear Independence and Linear Dependence
In Section 3.3 we saw that if a system of linear equations has more unknowns than
equations then there is more than one solution, provided there is a solution. If th
3.3 More Unknowns than Equations
Frequently one has a system of linear equations with more unknowns than equations. As
we shall see, when this occurs there is more than one solution. Here is an example.
Example 1. In the context of Example 1 of section 3.
4.3 Leontif Input-Output Models
These are models of the economy or portions of the economy. In these models a portion
of the economy is divided into sectors. Let's look a greatly simplified example.
Example 1. Suppose we look at the following sectors of t
4 Inverses of Matrices
This chapter is concerned with inverses of matrices. In a sense this is just another aspect
of solving linear equations. However, since inverses of matrices are so important, we
devote a separate chapter to this topic.
4.1 Inverses
3.6
Existence of solutions to linear equations
So far most of the systems of equations that we have encountered have had solutions.
Now we want to look at situations where the system of equations does not have a
solution. In particular, we want to give a