Overview

• Definitions• Basic matrix operations (+, -, x)• Determinants and inverses

Some Definitions …

• Zero Matrix

• Identity Matrix

• Diagonal Matrix

I A = A I = A

Basic Operations

• Addition, Subtraction, Multiplication

hdgc

fbea

hg

fe

dc

ba

hdgc

fbea

hg

fe

dc

ba

dhcfdgce

bhafbgae

hg

fe

dc

ba

Just add elements

Just subtract elements

Multiply each row by each column

Multiplication

• Is AB = BA? Maybe, but maybe not!

• Is multiplication commutative?

......

...bgae

hg

fe

dc

ba

......

...fcea

dc

ba

hg

fe

42

31A

53

12B

Try for the 2 matrices below

42

31A

53

12B

2216

1611

54123422

53113321

53

12

42

31BAC

Multiplication

Is AB = BA?

2913

104

45332513

41322112

42

31

53

12ABD

Multiplication is NOT commutative AB = BA

Inverse of a Matrix

• Identity matrix: AI = A

• Some matrices have an inverse, such that:AA-1 = I

100

010

001

I

Inverse of a 2x2 Matrix

Matrix Inverse (Intro)

A A-1= A-1 A = I

I

100

010

001

71

00

051

0

0031

700

050

003

DD 1

Properties

A-1 only exists if A is square (n x n)

Determinant of a 2x2 Matrix

• The determinant of the matrix A is denoted |A|.

• Matrix A has no inverse whenever |A|= 0.• A matrix with no inverse is SINGULAR.

42

31AE.g.

, so an inverse exists

26

13B

, so no inverse exists

Inverse of a 2x2 Matrix

• AA-1 = I• If = 0, then A has no inverse

– A is SINGULAR

dc

baA

42

31AE.g.

12

34

2

11A

5.01

5.12

Inverse of a 2x2 Matrix

• AA-1 = I• If |A| = 0, then A has no inverse

– A is SINGULAR

dc

baA

5.01

5.12

2344

5.15.132

10

01 The 2x2identity matrix