review of matrices or a fast introduction. why study matrices? fundamental tools for linear systems

Review of MatricesOr A Fast Introduction

Why Study Matrices?Fundamental Tools For Linear

Systems

Matrices Come From Systems of Equations

We’ve known how to find

For different right hand sides

Matrices Come From Systems of Equations

Matrices are all about what happens with different

Matrix

Vector

Matrices Come From Systems of Equations

Matrices are all about what happens with different

Matrices Come From Systems of Equations

Fundamental Definition:Matrix Vector

Multiplication

(Think of it as a system of equations!)Try

it!

Matrices Come From Systems of Equations

Fundamental Definition:Matrix Vector

Multiplication

(Think of it as a system of equations!)Try

it!

Matrices Come From Systems of Equations

Fundamental Definition:Matrix Vector

Multiplication

Notation

First IndexIs Row

Second Index is Column

Rows

Columns

Matrices Come From Systems of Equations

Fundamental Definition:Matrix Vector

Multiplication

Notation

Second ColumnFirst

Row

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

In General

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

In General

Matrices Come From Systems of Equations

Other Important DefinitionsAdditio

n

In General

Matrices Come From Systems of Equations

Other Important DefinitionsScalar

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsAddition and Scalar Multiplication Give Subtraction

Without a Vector

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Matrices Come From Systems of Equations

Other Important DefinitionsMatrix

Multiplication

Special MatricesIdentit

y

Zero

Matrix Algebra RulesA Lot Like Normal Algebra

Rules

Notice:

(Addition Commutes)(Addition Associates)(Multiplication

Associates)(Distribute)

(Scalars Distribute)

(Multiplication

DOES NOTCommute)

Integration and Differentiation

Integration and Differentiation

Matrix InversionMany Matrices Have an

Inverse

Inverse Has Special Properties

For Example:

Inverse FormulaMatrix Inverse For 2

by 2

The Determinant

Inverse Exists If and Only If

Inverse Helps Solve Systems of Equations

Inverse Helps Solve Systems of Equations

Inverse Helps Solve Systems of Equations

Inverse Helps Solve Systems of Equations

Inverse Helps Solve Systems of Equations

Inverse Helps Solve Systems of Equations

Singular MatricesMatrix Inverse For 2

by 2

If the Determinant is 0

The Inverse Does Not Exist!

May Not Have A Solution

Example

Determinant is 0

Impossible!

Really Important Fact

If (and only if) the Determinant is 0

There is a Vector

Where

Example

Three Ways of Saying The Same Thing

(1) The Determinant is 0

(2) The Matrix Has No Inverse (Is Singular)(3) There Is A Vector

Where

Summary

• Matrices Come From Systems Of Equations

• We can do Algebra on Matrices

• Matrices are called “singular” if the determinant is 0. Such matrices have no inverses.

Questions?