Matrices - Ch. 1.6

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Section Two: Matrices

Textbook: Ch. 1.6

GOALS OF THIS CHAPTER

- define matrix multiplication

- figure out when matrix multiplication fails

- matrix multiplication as linear combinations

- see properties of sigma notation

- see properties of matrix multiplication

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MULTIPLICATION OF MATRICES

If A is an m x p matrix and B is a p x n matrix, then we can do thefollowing preliminary steps:

Matrix multiplication is NOTHING like

regular multiplication!

Bp x n

STEP ONE: Check to see if the number of columns of thefirst matrix is equal to the number of rows of the secondmatrix. If not, the multiplication doesnt exist.

Am x p

Since these numbers are equal, wemove onto step two.

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If A is an m x p matrix and B is a p x n matrix, then we can do thefollowing preliminary steps:

Matrix multiplication is NOTHING like

regular multiplication!

Bp x n

STEP TWO: The other two letters or numbers will tell youthe size of the new matrix AB.

Am x p

Read off these letters from left toright. The matrix AB is m x n.

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If steps one and two make sense, then we calculate the entries of ABby doing a summation. If we label the entries of AB, A and B as abij,aij, bij, respectively, then:

a ikab ij = bkjk = 1

p

This just means that the (i,j) entry of AB uses allthe information from row i of A and all theinformation from column j of B. We will revisit this

again later.

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Ex. 1 Row and column matrices


These couple calculations are the mostimportant. I will show you how to break biggermultiplications down into these ones.

B =

-1

0

10

C =3 -5 01

-1A =

C3 x 1B1 x 3Calculate BC:

C3 x 1B1 x 3 BC is 1 x 1

BC exists

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BC =-1

0

10

3 -5 0

= 3*(-1) +

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BC =-1

0

10

3 -5 0

= 3(-1) + (-5)*0 +

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BC =-1

0

10

3 -5 0

= 3(-1) + (-5)0+ 0*(10)

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BC =-1

0

10

3 -5 0

= 3(-1) + (-5)0+ 0(10)

= -3

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B =

-1

010

C =3 -5 01-1

A =

Calculate AB:

B1 x 3A2 x 1

B1 x 3A2 x 1 AB is 2 x 3

AB exists

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AB =

=

3 -5 01

-1

Recall that the (1,1) entry ofAB comes from all theinformation in row one of Aand column one of B.

= 1*3

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AB =

=

3 -5 01

-1

1*3 1*(-5)

The (1,2) entry of AB comesfrom all the information inrow one of A and column two

of B.

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AB =

=

3 -5 01

-1

1*3 1*(-5) 1*0

The (1,3) entry of AB comesfrom all the information inrow one of A and column

three of B.

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AB =

=

3 -5 01

-1

1*3 1*(-5) 1*0

(-1)*3

The (2,1) entry of AB comesfrom all the information inrow two of A and column one

of B.

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AB =

=

3 -5 01

-1

1*3 1*(-5) 1*0

(-1)*3 -1*(-5)

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AB =

=

3 -5 01

-1

1*3 1*(-5) 1*0

(-1)*3 -1*(-5) (-1)*0

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AB =

=

3 -5 01

-1

3 -5 0

-3 5 0

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B =

-1

010

C =3 -5 01-1

A =

Calculate AC:

C3 x 1A2 x 1

These numbers are not equal, so we say AC is not

defined or AC does not exist.

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Ex. 2 Some larger matrices


Next we will break down a larger multiplicationinto a row/column matrix multiplication.

B =A =

Calculate AB:

B3 x 2A2 x 3

1 2 -13 1 4

-2 5

4 -3

2 1

B3 x 2A2 x 3 AB is 2 x 2

AB exists

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AB = 1 2 -1

3 1 4

-2 5

4 -3

2 1

1 2 -1 -24

2=

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AB = 1 2 -13 1 4

-2 5

4 -3

2 1

1 2 -1 1 2 -1-24

2

5-3

1=

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AB = 1 2 -13 1 4

-2 5

4 -3

2 1

1 2 -1 1 2 -1

3 1 4

-24

2

-2

4

2

5-3

1=

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AB = 1 2 -13 1 4

-2 5

4 -3

2 1

1 2 -1 1 2 -1

3 1 4 3 1 4

-24

2

-2

4

2

5

-3

1

5-3

1=

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1(-2)+2*4+(-1)2 1*5+2(-3)+(-1)1

3(-2)+1*4+4*2 3*5+1(-3)+4*1AB =

4 -2

6 16=

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THE IDENTITY MATRIX

Recall that the identity matrix is the square matrix

with only ones on the main diagonal.

When we multiply numbers, any number multiplied by the number 1 isjust itself.

I1 0

0 1=

When we multiply a matrix by an identity matrix ofthe right size, we get the matrix we started with(provided the multiplication makes sense)!

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THE IDENTITY MATRIX

Lets say that I is a 2x2 matrix, A is a 2x3 matrix

and B is a 3x2 matrix.

When multiplying with I, all you really have to do is check that themultiplication makes sense:

I1 0

0 1= A =

1 2 -1

3 1 4B =

-2 5

4 -3

2 1

Calculate AI:

I2 x 2A2 x 3

AI does not exist

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THE IDENTITY MATRIX

Lets say that I is a 2x2 matrix, A is a 2x3 matrix

and B is a 3x2 matrix.

What about the other way around?

I1 0

0 1= A =

1 2 -1

3 1 4B =

-2 5

4 -3

2 1

Calculate IA:

A2 x 3I2 x 2

A2 x 3I2 x 2 IA is 2 x 3

IA exists

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THE IDENTITY MATRIX

Lets say that I is a 2x2 matrix, A is a 2x3 matrixand B is a 3x2 matrix.

In fact, IA is just equal to A (you should check this)! So provided themultiplication makes sense, multiplying by I is pretty easy.

I1 0

0 1= A =

1 2 -1

3 1 4B =

-2 5

4 -3

2 1

I A = A

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THE IDENTITY MATRIX

Lets say that I is a 2x2 matrix, A is a 2x3 matrixand B is a 3x2 matrix.

What side can we multiply I on when dealing with the matrix B?

I1 0

0 1= A =

1 2 -1

3 1 4B =

-2 5

4 -3

2 1

IB = BBI = B

Since B is 3x2, we mustmultiply by I on the rightfor the multiplication tomake sense!

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MATRIX MULTIPLICATION AS LINEAR

COMBINATIONS

a11x1 + a12x2+ + a1nxn

a21x1 + a22x2+ + a2nxn

am1x1 + am2x2+ + amnxn

x1

x2

xn

a11 a12 a1n

a21 a22 a2n

am1 am2 amn

=

Let A be an mxn matrix and let x be an n-vector (or an nx1 columnmatrix). Consider the multiplication Ax:

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COMBINATIONS

a11x1 + a12x2+ + a1nxn

a21x1 + a22x2+ + a2nxn

am1x1 + am2x2+ + amnxn

=

We can rewrite the right-hand side of the equality as

a11

a21

am1

a12

a22

am2

a1n

a2n

amn

+ +x1 x2 xn+

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COMBINATIONS

=

By re-labeling the n-vectors as c1, c2, , cn (the subscript denotesthe column of A), we express the multiplication Axas a linearcombination of the column vectors of the matrix A.

a11

a21

am1

a12

a22

am2

a1n

a2n

amn

+ +x1 x2 xn+Ax =x1c1 + x2c2++ xncn

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COMBINATIONS

We have shown that the product of an mxn matrix A and a columnvector x is a linear combination of the columns of A.

Linear combinations play an important role in higher linear algebracourses and are a fundamental tool in digital music.

We could generalize this finding to a matrix Athat is mxn and a matrix B that is nxp. Theproduct AB is a linear combination of the columnsof A as well! (see page 69-70 of the text formore info)

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COMBINATIONS

=

Ex. 4 - Lets see an example of this:

1 0 -3

-1 2 57 10 4

-1

0

10

-31

51

93

-31

51

93

=

1-1

7

02

10

-35

4

+ + 100-1

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SIGMA NOTATION

Let A be an mxn matrix, and let B be an nxp matrix. We have shownthat the product AB is defined and is of size mxp. To figure out the(i,j) entry of the matrix AB, we could use the formula:

a ikab ij = bkjk = 1

p

We call this sigma notation, but how exactly doesit work?

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SIGMA NOTATION

Lets say we want the (1,1) entry of AB. This means we

fill in i=1 and j=1 in our formula:

a1kab11 = bk1k = 1

p

Next, we let the variable k=1 to start, then we add the term k=2, k=3,all the way up to k=p:

= a11b11 + a12b21+ + a1pbp1

k=1 k=2 k=p. . .

ab11

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SIGMA NOTATION

Ex. 6 Sum of the first five squares

- done on overhead

Thm. 7 Facts about sigma notation

- done on overhead

- no proof required

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PROPERTIES OF MATRIX MULTIPLICATION

Thm. 8 Properties of matrix multiplication

Provided that A, B and C are matrices of the right size formultiplication, and that r is any number, we have the following:

(1) A(BC) = (AB)C

(2) A(B+C) = AB+AC

(3) (A+B)C = AC+BC

(4) A(rB) = r(AB)

(5) (AB)

T

= B

T

A

T

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PROPERTIES OF MATRIX MULTIPLICATION

Thm. 8 Properties of matrix multiplication

Proof Done on overhead

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Matrices - Ch. 1.6