Download - MAT 2401 Linear Algebra

MAT 2401Linear Algebra

2.1 Operations with Matrices

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HW...

If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE.

GJE is the corner stone of this class, you really need to figure it out.

Today

Written HW Again, today may be longer. It is

more efficient to bundle together some materials from 2.2.

Next class session will be shorter.

Preview

Look at the algebraic operations of matrices

“term-by-term” operations•Matrix Addition and Subtraction

•Scalar Multiplication Non-“term-by-term” operations

•Matrix Multiplication

Matrix

If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn.

1 2

1

2

n

m

Notations

Matrix

th

t

h

ij

ij

j

A a

ai

Notations

Matrix Example:

11

23

1 1 1 4

2 2 5 11

4 6 8 24

A

a

a

th

t

h

ij

ij

j

A a

ai

Special Cases

Row Vector

Column Vector

1 2 nb b b

1

2

m

c

c

c

Matrix Addition and Subtraction

Let A = [aij] and B = [bij] be mxn matrices

Sum: A + B = [aij+bij]

Difference: A-B = [aij-bij]

(Term-by term operations)

Example 1

1 2

3 1

0 2

3 2

A

B

A B

A B

Scalar Multiplication

Let A = [aij] be a mxn matrix and c a scalar.

Scalar Product: cA=[caij]

Example 2

1 2

3 1A

2A

Matrix Multiplication

Define multiplications between 2 matrices

Not “term-by-term” operations

Motivation

2 3 4 5x y z

The LHS of the linear equation consists of two pieces of information:•coefficients: 2, -3, and 4

•variables: x, y, and z

Motivation

2 3 4 5

2 3 4 5

x y z

x

y

z

Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors.

Row-Column Product

1

21 2 1 1 2 2n n n

n

b

ba a a a b a b a b

b

same no. of elements

Example 3

2

21 3 2 4

1

2

Matrix Multiplication

1

21

11 12 111 1

21 22

2

2

1

1 1 2

1

j

ji i ip

pj

i j i j

pn

n

p pnm m mp

ip pj

b

ba a a

b

a b a

a a ab b

b b

b b

b a

a

b

a a

th

th ijc

j

i

Example 4

1 2 0 1

1 0 1 0

Example 5 (a)

4 21 2 1

0 12 3 1

2 1

Scratch:Q: Is it possible to multiply the 2 matrices?

Q: What is the dimension of the resulting matrix?

Example 5 (b)

1 2 3 2

2 3 1 3



Example 5 (c)

11 2

1



Example 5 (d)

11 2

1



Remark: 11 2 ,

1A B

Example 5 (e)

1 1 1 1

1 1 1 1



Remark:1 1 1 1

, 1 1 1 1

A B

Example 5 (f)

1 0 1 2

0 1 3 4



Remark:1 0 1 2

, 0 1 3 4

I A

Interesting Facts

The product of mxp and pxn matrices is a mxn matrix.

In general, AB and BA are not the same even if both products are defined.

AB=0 does not necessary imply A=0 or B=0.

Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity.

Identity Matrix

nxn Square Matrix

1 0 0

0 1

0

0 0 1

nI I

Zero Matrix

mxn Matrix with all zero entries

0 0 0

0 00 0

0

0 0 0

mn

Representation of Linear System by Matrix Multiplication

4

2 2 5 11

4 6 8 24

x y z

x y z

x y z


2 2 5 11

4

4

4 6 8 2

x y z

z

x y z

x y


4

4 6 8

11

2

2 5

4

2x y

y

z

x z

x y z

Let

Then the linear system is given by


4

2 2 5 11

4 6 8 24

x y z

x y z

x y z

1 1 1 4

2 2 5 , , 11

4 6 8 24

x

A X y b

z

Let

Then the linear system is given by

Remark

It would be nice if “division” can be defined such that:

(2.3) Inverse

1 1 1 4

2 2 5 , , 11

4 6 8 24

x

A X y b

z

HW...

If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE.

GJE is the corner stone of this class, you really need to figure it out.

Download - MAT 2401 Linear Algebra

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