Matrices III: Matrix Algebra &

Automatic-Matrices!

School of Economics, UNSW

2011

Contents

1 Introduction

1

2 More on Matrix Algebra

2.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2

3 Matrices on Computers

4

1

Introduction

Having learnt a bit about what is going on with matrices, we just have a few more loose ends to tie up. Speciﬁcally, we’ll go over the algebra of matrices. This is just like what we would normally do with ‘scalar’ algebra: rearranging equations, simplifying, making a certain pro-numeral the subject of the equation. However, with matrices, there are some very important diﬀerences in the rules and how to apply them. Pay attention.

Following this, we’ll cover a couple of important pieces of terminology – singular and non-singular on the one hand and consistent and inconsistent on the other.

These are really just labels for concepts that we are already familiar with, but they have especial relevance when we move to using computers. I say this, because although

Microsoft Excel (which we’ll be using for the moment) doesn’t have very elaborate error-messages, other programs do. For instance, Matlab might report (when trying to obtain A−1 ),

>> inv(A)

Warning: Matrix is close to singular or badly scaled.

Results may be inaccurate.

...

what does this mean? See below.

The main purpose of the second part of this lecture is to introduce the way of matrices on computers, speciﬁcally using Microsoft Excel. As with most computer packages,

Excel is very fast to do lots of things, but fast doesn’t always equal correct! It’s for this reason that we have gone through matrices ‘by hand’ to a point to begin with, so that we can actually uncover when the computer is telling us ﬁbs!

1

ECON 1202/ECON 2291: QABE

c School of Economics, UNSW

Agenda

1. A little more on Matrix Algebra.

2. Using computers to do Matrix Maths:

• Small operations;

• Linear equations;

• Big operations.

2

More on Matrix Algebra

HPW

6.1-6.3

Deﬁnition | Properties of the inverse

Some useful properties of the inverse,

• If the inverse of the square matrix A exists, we call A non-singular, otherwise we call A singular;

• If A−1 exists, it is unique;

• Other properties,

(AB)−1 = B−1 A−1

(A−1 )−1 = A

(A )−1 = (A−1 )

(A + B)−1 = A−1 + B−1

(normally)

Example:

Suppose A, B and C are all invertible matrices, and

[C−1 A + X(A−1 B)−1 ]−1 = C , express X in terms of A, B and C.

2.1

Consistency

What if there isn’t a solution?

• We have taken for granted so far that our systems of linear equations will be solvable, however ...

• For now, we notice that,

QABE Lecture 7

2

ECON 1202/ECON 2291: QABE

c School of Economics, UNSW

Deﬁnition | Consistency

A linear system having no solutions is said to be inconsistent, whereas a system having either one or more solutions is said to be consistent.

Example:

Determine whether the following linear system

⎡

1 −3 5

[A|b] = ⎣ 0 1 2

0 0 0

is consistent,

⎤

3

2 ⎦.

1

A relationship to the Determinant??

Example:

Determine whether the following linear system determinant of A,

⎡

1 −3 5

⎣ 0 1 2

[A|b] =

0 0 0

is consistent by checking the

⎤

3

2 ⎦ .

1

Independence, Matrix style

• Whenever we have the case that one row or column is not independent of another row or column (respectively) in our matrix, we will ﬁnd the inverse (and so, a solution) hard to come by.

• This is the principle of linear independence... if for some reason you ﬁnd that a matrix is singular, then you should look at the relationship…