Chapter 3 Matrices - 3.2 Matrix Algebrabtravers.weebly.com/.../7/2/9/6729909/3.2_matrix_algebra.pdfChapter 3 Matrices Movies 3.2 Matrix Algebra 1 Matrix Multiplication 2 Properties

1 MatrixMultiplication

2 Properties toNote

3 ElementaryMatrices

4 Math in theMoviesChapter 3 Matrices

3.2 Matrix Algebra


2 Properties toNote


4 Math in theMovies

Matrix Multiplication

Example

Foxboro Stadium has three main concession stands, locatedbehind the south, north and west stands. The top-sellingitems are peanuts, hot dogs and soda. Sales for the seasonopener are recorded in the first matrix below, and the prices(in dollars) of the three items are given in the second matrix.

SouthNorthWest

120 250 305207 140 41939 120 190

2.003.002.75

PeanutsHot Dogs

Soda

How can we find the total sales from the south stands?

120(2.00) + 250(3.00) + 305(2.75) = $1828.75


2 Properties toNote


4 Math in theMovies


Example


SouthNorthWest

120 250 305207 140 41939 120 190

2.003.002.75

PeanutsHot Dogs

Soda


120(2.00) + 250(3.00) + 305(2.75) = $1828.75


2 Properties toNote


4 Math in theMovies


Example


SouthNorthWest

120 250 305207 140 41939 120 190

2.003.002.75

PeanutsHot Dogs

Soda


120(2.00) + 250(3.00) + 305(2.75) = $1828.75


2 Properties toNote


4 Math in theMovies


Similarly, for the north and west stands, respectively, we get

207(2.00) + 140(3.00) + 419(2.75) = $1986.25

and

39(2.00) + 120(3.00) + 190(2.75) = $940.50

We can arrive at this, using matrix multiplication, where thesystem would look like120 250 305

207 140 41939 120 190

2.003.002.75

=

1828.751986.25940.50


2 Properties toNote


4 Math in theMovies


Similarly, for the north and west stands, respectively, we get

207(2.00) + 140(3.00) + 419(2.75) = $1986.25

and

39(2.00) + 120(3.00) + 190(2.75) = $940.50

We can arrive at this, using matrix multiplication, where thesystem would look like120 250 305

207 140 41939 120 190

2.003.002.75

=

1828.751986.25940.50


2 Properties toNote


4 Math in theMovies


DefinitionIf A = [aij ] is an m × n matrix and B = [bij ] is an n × pmatrix, then the product AB is an m × p matrix

AB = [cij ]

where

cij =n∑

k=1

aikbkj = ai1b1j + ai2b2j + . . . + ainbnj

This is a fancy way of saying that the i , j position in theanswer matrix is the dot product of the i th row of the firstmatrix and the j th column of the second matrix.We also have to make sure that the sizes of the matrices areappropriate for multiplying matrices.


2 Properties toNote


4 Math in theMovies



AB = [cij ]

where

cij =n∑

k=1


This is a fancy way of saying that the i , j position in theanswer matrix is the dot product of the i th row of the firstmatrix and the j th column of the second matrix.

We also have to make sure that the sizes of the matrices areappropriate for multiplying matrices.


2 Properties toNote


4 Math in theMovies



AB = [cij ]

where

cij =n∑

k=1


This is a fancy way of saying that the i , j position in theanswer matrix is the dot product of the i th row of the firstmatrix and the j th column of the second matrix.We also have to make sure that the sizes of the matrices areappropriate for multiplying matrices.


2 Properties toNote


4 Math in theMovies

Matrix Multiplication Example

Example

Find the product AB, where

A =

−1 34 −25 0

and B =

[−3 2−4 1

]

−1 34 −25 0

[−3 2−4 1

]=

c11 c12c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


Example

Find the product AB, where

A =

−1 34 −25 0

and B =

[−3 2−4 1

]

−1 34 −25 0

[−3 2−4 1

]=

c11 c12c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

c11 c12c21 c22c31 c32

c11 = (−1)(−3) + 3(−4) = −9

−1 34 −25 0

[−3 2−4 1

]=

−9 c12c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

c11 c12c21 c22c31 c32

c11 = (−1)(−3) + 3(−4) = −9

−1 34 −25 0

[−3 2−4 1

]=

−9 c12c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

c11 c12c21 c22c31 c32

c11 = (−1)(−3) + 3(−4) = −9

−1 34 −25 0

[−3 2−4 1

]=

−9 c12c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

−9 c12c21 c22c31 c32

c12 = −1(2) + 3(1) = 1

−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

−9 c12c21 c22c31 c32

c12 = −1(2) + 3(1) = 1

−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

−9 c12c21 c22c31 c32

c12 = −1(2) + 3(1) = 1

−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22c31 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22c31 c32

c31 = 5(−3) + 0(−4) = −15

−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22−15 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22c31 c32

c31 = 5(−3) + 0(−4) = −15

−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22−15 c32


2 Properties toNote


4 Math in theMovies


−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22c31 c32

c31 = 5(−3) + 0(−4) = −15

−1 34 −25 0

[−3 2−4 1

]=

−9 1c21 c22−15 c32


2 Properties toNote


4 Math in theMovies


Continuing, we get ...−1 34 −25 0

[−3 2−4 1

]=

−9 1−4 6−15 10


2 Properties toNote


4 Math in theMovies


Example

[1 0 32 −1 −2

]−2 4 21 0 0−1 1 −1

=

[−5 7 −1−3 6 6

]

Example [3 4−2 5

] [1 00 1

]=

[3 4−2 5

]

What do we call the second matrix?


2 Properties toNote


4 Math in theMovies


Example

[1 0 32 −1 −2

]−2 4 21 0 0−1 1 −1

=

[−5 7 −1−3 6 6

]

Example [3 4−2 5

] [1 00 1

]=

[3 4−2 5

]



2 Properties toNote


4 Math in theMovies


Example

[1 0 32 −1 −2

]−2 4 21 0 0−1 1 −1

=

[−5 7 −1−3 6 6

]

Example [3 4−2 5

] [1 00 1

]=

[3 4−2 5

]



2 Properties toNote


4 Math in theMovies


Example

[1 0 32 −1 −2

]−2 4 21 0 0−1 1 −1

=

[−5 7 −1−3 6 6

]

Example [3 4−2 5

] [1 00 1

]=

[3 4−2 5

]



2 Properties toNote


4 Math in theMovies


Example

[1 0 32 −1 −2

]−2 4 21 0 0−1 1 −1

=

[−5 7 −1−3 6 6

]

Example [3 4−2 5

] [1 00 1

]=

[3 4−2 5

]



2 Properties toNote


4 Math in theMovies


Example [1 21 1

] [−1 21 −1

]=

[1 00 1

]

What is the relationship between these two matrices?

Example

[1 −2 −3

] 2−11

=[1]


2 Properties toNote


4 Math in theMovies


Example [1 21 1

] [−1 21 −1

]=

[1 00 1

]


Example

[1 −2 −3

] 2−11

=[1]


2 Properties toNote


4 Math in theMovies


Example [1 21 1

] [−1 21 −1

]=

[1 00 1

]


Example

[1 −2 −3

] 2−11

=[1]


2 Properties toNote


4 Math in theMovies


Example [1 21 1

] [−1 21 −1

]=

[1 00 1

]


Example

[1 −2 −3

] 2−11

=

[1]


2 Properties toNote


4 Math in theMovies


Example [1 21 1

] [−1 21 −1

]=

[1 00 1

]


Example

[1 −2 −3

] 2−11

=[1]


2 Properties toNote


4 Math in theMovies


Example 2−11

[1 −2 −3]

=

2 −4 −6−1 2 31 −2 −3


2 Properties toNote


4 Math in theMovies


Example 2−11

[1 −2 −3]

=

2 −4 −6−1 2 31 −2 −3


2 Properties toNote


4 Math in theMovies

Commutativity (and the lack of)

Note: Matrices are not necessarily commutative. Thinkabout the size of the matrices ...

Even if the sizes work, there is no guarantee that there willbe equality.

Example

Find the product AB and BA if

A =

[1 2−2 3

]and B =

[1 −1−3 5

]AB =

[−5 9−11 17

]but BA =

[3 −1−13 9

]


2 Properties toNote


4 Math in theMovies




Example


A =

[1 2−2 3

]and B =

[1 −1−3 5

]AB =

[−5 9−11 17

]but BA =

[3 −1−13 9

]


2 Properties toNote


4 Math in theMovies




Example


A =

[1 2−2 3

]and B =

[1 −1−3 5

]

AB =

[−5 9−11 17

]but BA =

[3 −1−13 9

]


2 Properties toNote


4 Math in theMovies




Example


A =

[1 2−2 3

]and B =

[1 −1−3 5

]AB =

[−5 9−11 17

]

but BA =

[3 −1−13 9

]


2 Properties toNote


4 Math in theMovies




Example


A =

[1 2−2 3

]and B =

[1 −1−3 5

]AB =

[−5 9−11 17

]but BA =

[3 −1−13 9

]


2 Properties toNote


4 Math in theMovies

The Identity MatrixWe saw two instances of the identity matrix in the priorexamples:

1. The product of a matrix and the identity is the originalmatrix

2. The product of a matrix and it’s inverse is the identitymatrix

We actually have two identity matrices, depending on theoperation.

Additive Identity

For any matrix A ∈Mmn, the matrix 0mn is theadditive identity and has the propertyA + 0mn = A = 0mn + A.

We generally refer to this as the zero matrix rather than anidentity matrix.


2 Properties toNote


4 Math in theMovies





Additive Identity




2 Properties toNote


4 Math in theMovies





Additive Identity




2 Properties toNote


4 Math in theMovies





Additive Identity




2 Properties toNote


4 Math in theMovies





Additive Identity




2 Properties toNote


4 Math in theMovies





Additive Identity




2 Properties toNote


4 Math in theMovies

The Identity Matrix

Multiplicative Identity

For any matrix A ∈Mn, the matrix In is themultiplicative identity and has the property AIn = A = InA.

In =

1 0 0 . . . 00 1 0 . . . 0...

......

......

0 0 0 . . . 1


2 Properties toNote


4 Math in theMovies

The Identity Matrix

Multiplicative Identity

For any matrix A ∈Mn, the matrix In is themultiplicative identity and has the property AIn = A = InA.

In =

1 0 0 . . . 00 1 0 . . . 0...

......

......

0 0 0 . . . 1


2 Properties toNote


4 Math in theMovies

The transpose of a matrix

DefinitionThe transpose of a matrix, denoted AT , is the matrix formedfrom the matrix A = [aij ] by interchanging the rows and thecolumns. AT = [aji ].

Visually speaking, the transpose of a matrix is a reflectionover the main diagonal.

A =

[1 2 34 5 6

]

AT =

1 42 53 6


2 Properties toNote


4 Math in theMovies




A =

[1 2 34 5 6

]

AT =

1 42 53 6


2 Properties toNote


4 Math in theMovies




A =

[1 2 34 5 6

]

AT =

1 42 53 6


2 Properties toNote


4 Math in theMovies




A =

[1 2 34 5 6

]

AT =

1 42 53 6


2 Properties toNote


4 Math in theMovies

Properties of Transposes

TheoremIf A and B are matrices (with sizes such that the givenmatrix operations are defined) and c is a scalar, then thefollowing properties are true:

1.(AT)T

= A

2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT

Why do these properties hold?


2 Properties toNote


4 Math in theMovies



1.(AT)T

= A

2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT



2 Properties toNote


4 Math in theMovies



1.(AT)T

= A

2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT



2 Properties toNote


4 Math in theMovies



1.(AT)T

= A

2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT



2 Properties toNote


4 Math in theMovies



1.(AT)T

= A

2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT



2 Properties toNote


4 Math in theMovies



1.(AT)T

= A

2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT



2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT

If A is an a× b matrix, then B must be a b × c matrix.

Then, the product AB is an a× c matrix, so (AB)T is ac × a matrix. Furthermore, AT is a b× a matrix and BT is ac × b matrix, so ATBT cannot exist, but BTAT is a c × amatrix.

Since (AB)T and BTAT have the same size, we now needto show that the entries are the same. First, note that thei , j th entry of (AB)T is the same as the j , i th entry of AB.

Now, the i , j th entry of BTAT is the dot product of the i th

row of BT , which is the i th column of B, and the j th columnof AT , which is the j th row of A. That is, the i , j th entry ofBTAT is the dot product of the j th row of A and i th columnof B.


2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT

If A is an a× b matrix, then B must be a b × c matrix.Then, the product AB is an

a× c matrix, so (AB)T is ac × a matrix. Furthermore, AT is a b× a matrix and BT is ac × b matrix, so ATBT cannot exist, but BTAT is a c × amatrix.





2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT

If A is an a× b matrix, then B must be a b × c matrix.Then, the product AB is an a× c matrix, so (AB)T is ac × a matrix.

Furthermore, AT is a b× a matrix and BT is ac × b matrix, so ATBT cannot exist, but BTAT is a c × amatrix.





2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT

If A is an a× b matrix, then B must be a b × c matrix.Then, the product AB is an a× c matrix, so (AB)T is ac × a matrix. Furthermore, AT is a b× a matrix and BT is ac × b matrix, so ATBT cannot exist,

but BTAT is a c × amatrix.





2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT

If A is an a× b matrix, then B must be a b × c matrix.Then, the product AB is an a× c matrix, so (AB)T is ac × a matrix. Furthermore, AT is a b× a matrix and BT is ac × b matrix, so ATBT cannot exist, but BTAT is a

c × amatrix.





2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT

If A is an a× b matrix, then B must be a b × c matrix.Then, the product AB is an a× c matrix, so (AB)T is ac × a matrix. Furthermore, AT is a b× a matrix and BT is ac × b matrix, so ATBT cannot exist, but BTAT is a c × amatrix.





2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT


Since (AB)T and BTAT have the same size, we now needto show that the entries are the same.

First, note that thei , j th entry of (AB)T is the same as the j , i th entry of AB.




2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT






2 Properties toNote


4 Math in theMovies

Proof of part 4

(AB)T = BTAT






2 Properties toNote


4 Math in theMovies

Symmetric Matrices

DefinitionA matrix is said to be symmetric if A = AT . Then aij = ajifor all i 6= j

This implies that all symmetric matrices must be square.


2 Properties toNote


4 Math in theMovies

Symmetric Matrices

DefinitionA matrix is said to be symmetric if A = AT . Then aij = ajifor all i 6= j

This implies that all symmetric matrices must be square.


2 Properties toNote


4 Math in theMovies

Elementary Matrices

DefinitionAn elementary matrix Ei is a matrix that can be obtainedfrom the appropriately sized identity matrix by performingone row operation.

By using a sequence of elementary matrices, we candecompose a matrix.


2 Properties toNote


4 Math in theMovies

Elementary Matrices

DefinitionAn elementary matrix Ei is a matrix that can be obtainedfrom the appropriately sized identity matrix by performingone row operation.

By using a sequence of elementary matrices, we candecompose a matrix.


2 Properties toNote


4 Math in theMovies

Elementary Matrices

There are three types of elementary matrices:

1. Row Swap

E1 =

[0 11 0

]corresponds to R1 ⇔ R2

2. Multiply a row by a nonzero constant

E2 =

[k 00 1

]corresponds to kR1

3. Add a nonzero multiple of one row to another

E3 =

[1 20 1

]corresponds to 2R2 + R1


2 Properties toNote


4 Math in theMovies

Elementary Matrices


1. Row Swap

E1 =

[0 11 0



E2 =

[k 00 1

]corresponds to kR1


E3 =

[1 20 1



2 Properties toNote


4 Math in theMovies

Elementary Matrices


1. Row Swap

E1 =

[0 11 0



E2 =

[k 00 1

]corresponds to kR1


E3 =

[1 20 1



2 Properties toNote


4 Math in theMovies

Elementary Matrices


1. Row Swap

E1 =

[0 11 0



E2 =

[k 00 1

]corresponds to kR1


E3 =

[1 20 1



2 Properties toNote


4 Math in theMovies

Inverses of Elementary Matrices

1. Row Swap: the inverse is

(E1)−1 =

[0 11 0


2. Multiply a row by a nonzero constant: the inverse is

(E2)−1 =

[1k 00 1

]corresponds to 1

kR1

3. Add a nonzero multiple of one row to another: theinverse is

(E3)−1 =

[1 −20 1

]corresponds to −2R2 + R1

We will talk more about elementary matrices later on.


2 Properties toNote


4 Math in theMovies



(E1)−1 =

[0 11 0



(E2)−1 =

[1k 00 1

]corresponds to 1

kR1


(E3)−1 =

[1 −20 1




2 Properties toNote


4 Math in theMovies



(E1)−1 =

[0 11 0



(E2)−1 =

[1k 00 1

]corresponds to 1

kR1


(E3)−1 =

[1 −20 1




2 Properties toNote


4 Math in theMovies



(E1)−1 =

[0 11 0



(E2)−1 =

[1k 00 1

]corresponds to 1

kR1


(E3)−1 =

[1 −20 1




2 Properties toNote


4 Math in theMovies



(E1)−1 =

[0 11 0



(E2)−1 =

[1k 00 1

]corresponds to 1

kR1


(E3)−1 =

[1 −20 1




2 Properties toNote


4 Math in theMovies



(E1)−1 =

[0 11 0



(E2)−1 =

[1k 00 1

]corresponds to 1

kR1


(E3)−1 =

[1 −20 1




2 Properties toNote


4 Math in theMovies



(E1)−1 =

[0 11 0



(E2)−1 =

[1k 00 1

]corresponds to 1

kR1


(E3)−1 =

[1 −20 1




2 Properties toNote


4 Math in theMovies

Good Will Hunting


2 Properties toNote


4 Math in theMovies

Good Will Hunting

•4

1• •2 •3

A =

0 1 0 11 0 2 10 2 0 01 1 0 0

A3 =

2 7 2 37 2 12 72 12 0 23 7 2 2


2 Properties toNote


4 Math in theMovies

Good Will Hunting

•4

1• •2 •3

A =

0 1 0 11 0 2 10 2 0 01 1 0 0

A3 =

2 7 2 37 2 12 72 12 0 23 7 2 2


2 Properties toNote


4 Math in theMovies

Good Will Hunting

•4

1• •2 •3

A =

0 1 0 11 0 2 10 2 0 01 1 0 0

A3 =

2 7 2 37 2 12 72 12 0 23 7 2 2

Chapter 3 Matrices - 3.2 Matrix Algebrabtravers.weebly.com/.../7/2/9/6729909/3.2_matrix_algebra.pdfChapter 3 Matrices Movies 3.2 Matrix Algebra 1 Matrix Multiplication 2 Properties

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