7.2 Matrix AlgebraBenn Fox

Hannah Weber

3 2 4

-1 0 8

Order of Matrices

Going vertically is called the column. The column is listed first. Going horizontally is called the row. The row is listed second.

For the one above. . .there are 2 rows, so 2 would be listed first. There are 3 columns, so 3 would be listed second. The answer is: 2 x 3

!RowC

olu

mn

Orders of Matrices Continued

5 0 4

5 6 4

-1 0 8

What is the order of the matrices?

This matrix has 3 rows.This matrix has 3 columns.Therefore, the answer would be:

3 x 3

3 C

olu

mn

s

3 Rows

8 6

9 4

23 6

7 89

45 35

4 8

41 68

2 C

olu

mn

s7 Rows

This matrix has 7 rows.This matrix has 2 columns.Therefore, the answer would be:

7 x 2

Order of Matrices Examples

What is the order of the matrices?

5 0 4 9 7 6

5 6 4 2 4 8Example 1

5 0 3 3 5 6Example 2

Order of Matrices Answers

5 0 4 9 7 6

5 6 4 2 4 8

5 0 3 3 5 6Answer 2

Answer 1

2 Rows

6 C

olu

mn

s

There are 2 rows.There are 6 columnsTherefore, the answer is:

2 x 6

There is 1 rowsThere are 6 columnsTherefore, the answer is:

1 x 61 Row 6

Colu

mn

s

a11 a12 a13 a1n

a21 a22 a23 a2n

am1 am2 am3 amn

Identifying the Element Specified

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

An m x n matrix is a rectangular array of m rows and n columns of real numbers. The first subscript number identifies the row it’s in. The second subscript number identifies which column it’s in.

-2 -6 -4 -5

4 7 9 8

9 7 6 5

Identifying the Element Specified

For an example: Identify the element specified for the following matrix.

a13

Because the first letter is 1, that means it’s in the first row.

-2 -6 -4 -5

4 7 9 8

9 7 6 5

Because the second letter is 3, that means it’s in the third column.

-2 -6 -4 -5

4 7 9 8

9 7 6 5

Identifying the Element Specified Continued

-2 -6 -4 -5

4 7 9 8

9 7 6 5

Because it’s in the 1st row, and the 3rd column, the answer would be:

-4

Identify the element specified for the following matrix.

a21

-2 -6

4 7

9 7

It’s in the 2nd row. It’s in the 1st

column.

Because it’s in the 2nd row, and the 1st column, the answer would be:

4

43 5 7 -4 8

5 4 322 0 3

5 45 3 25 6

3 543 8 4 55

Identifying the Element Specified Examples

8 9 7 -4

5 8 2 0

4 12 3 25

2 6 8 4

2 3 8 9

Identify the element specified for the following matrix:

a44

Example 1

Identify the element specified for the following matrix:

a15

Example 2

43 5 7 -4 8

5 4 322 0 3

5 45 3 25 6

3 543 8 4 55

Identifying the Element Specified Answers

8 9 7 -4

5 8 2 0

4 12 3 25

2 6 8 4

2 3 8 9

Identify the element specified for the following matrix:

a44

Answer 1

Identify the element specified for the following matrix:

a15

Answer 2

Because it’s in the 4th row.Because it’s in the 4th column.The answer would be:

4

Because it’s in the 1st row.Because it’s in the 4th column.The answer would be:

8

To add or subtract matrices they need to have: The same sized rows The same sized columns

Adding and Subtracting Matrices

=

7 + 6 = 13

Add or subtract the numbers in the matching positions.!

Adding and Subtracting Continued

2+5 = 7, 4+4=8, 8+9=176+15=21, 9+6=15, 12+18=30

=

[9 43 25 4 ] [0 7

4 11 3] =

9-0=9, 4-7=-33-4=-1, 2-1=15-1=4, 4-3=1

Adding and Subtracting Examples

[ 4 5 67 8 910 11 12] [3 6 9

2 4 61 5 10 ] ?=Example

1

Example 2

[5 8 153 12 14 ] [ 1 18 1

33 13 10 ]=?

Adding and Subtracting Answer

[ 4 5 67 8 910 11 12] [3 6 9

2 4 61 5 10 ]=Answer

1

4+3=7, 5+6=11, 6+9=157+2=9, 8+4=12, 9+6=1510+1=11, 11+5=16, 12+10=22

[ 7 11 159 12 1511 16 22]

Answer 2

[5 8 153 12 14 ] [ 1 18 1

33 13 10 ]=5-1=4, 8-18=-10, 15-1=143-33=-30, 12-13=-1, 14-10=4

[ 4 −10 14−30 −1 4 ]

Multiplying Matrices

Columns of the first matrix equals the number of rows in the second matrix

! !

[4 3 45 6 6 ]

2 x 3

[9 42 36 4 ]3 x 2

The same numbers, mean you can multiple.

The outer numbers show what dimensions the answer will be.

!

1. Make sure the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix.

2. Multiply the numbers of each row of the 1st matrix with the numbers of each column in the second matrix.

3. Add up the products from step 2.

How to Multiply

Multiplying Matrices

[1 2 30 1 −1] [1 0

2 10 −1]

2 x 33 x 2

1(1) + 2(2) + 3(0) = 51(0) + 2(1) + 3(-1) = -10(1) + 1(2) + -1(0) = 20(0) + 1(1) + -1(-1) = 2

Multiply each number from the first row in the 1st matrix with the 1st column of the 2nd matrix, then add them up.

Multiply each number from the first row in the 1st matrix with the 2nd column of the 2nd matrix, then add them up.

Repeat these steps with the 2nd row of the 1st matrix.

1.2.

3.

[5 −12 2 ]Answer =

Multiplying Matrices Continued

[1 02 10 −1] [1 2 3

0 1 −1]1(1) + 0(0) = 11(2) + 0(1) = 21(3) + 0(-1) = 32(1) + 1(0) = 22(2) + 1(1) = 52(3) + 1(-1) = 50(1) + -1(0) = 00(2) + -1(1) = -10(3) + -1(-1) = 1

[1 2 32 5 50 −1 1]

[ 0 2 −2−6 4 −6 ] [ 4 5

−2 06 −6 ]

0(4) + 2(-2) + -2(6) = -160(5) + 2(0) + -2(-6) = 12-6(4) + 4(-2) + -6(6) = -68-6(5) + 4(0) + -6(-6) = 6

[−16 12−68 6 ]

[1 2 30 1 −1]3 =[3 6 9

0 3 −3 ]You distribute the 3 to each of the numbers.

Multiplying Matrices Examples

Example 1

[−1 23 4 ] [−3 5 ]=?

Example 2

[0 0 10 1 01 0 0 ][

1 2 12 0 1−1 3 4 ] =?

Example 3 [3 4 6

4 5 75 6 8 ]4 =?

Multiplying Matrices Answers

Answer 1

[−1 23 4 ] [−3 5 ]=

Answer 2

[0 0 10 1 01 0 0 ][

1 2 12 0 1−1 3 4 ] =

Answer 3 [3 4 6

4 5 75 6 8 ]4 =

The first matrix is a 2 x 2. The second matrix is a 1 x 2. Because of this, the answer would be: Not Possible

0(1) + 0(2) + 1(-1) = -10(2) + 0(0) + 1(3) = 30(1) + 0(1) + 1(4) = 40(1) + 1(2) + 0(-1) = 20(2) + 1(0) + 0(3) = 00(1) + 1(1) + 0(4) = 11(1) + 0(2) + 0(-1) = 11(2) + 0(0) + 0(3) = 21(1) + 0(1) + 0(4) = 1

[−1 3 42 0 11 2 1 ]

[12 16 2416 20 2820 24 32]

At a zoo, kids ride a train for 25 cents. Adults ride it for $1. Senior citizens for 75 cents. On a given day: 1,400 paid a total of $740 for the rides. There were 250 more kids than all other riders. Find the total amount of children, adults, and senior citizens.

Matrix Applications

1st step: assign letters for each variable. x=children y=adults z=senior citizens

2nd step: set up equations..25x+y+.75z=740x+y+z=1400x-(y+z)=250

.25 for each kid, 1 dollar for each adult, .75 for each senior citizen.

All three of the variables = 1,400 total paid

250 more kids than all other riders.

Matrix Applications Continued 3rd Step: Plug into calculator as a matrix

4th Step: Find inverse of with the calculator

5th Step: Multiply the answer from the 4th step with

=

Answer: =

Matt has 74 coins: nickels, dimes, and quarters. For a total of $8.85. The number of nickels and quarters is 4 more than the number of dimes. Find the number of each coin.

Matrix Application Example

Example 1

Matt has 74 coins: nickels, dimes, and quarters. For a total of $8.85. The number of nickels and quarters is 4 more than the number of dimes. Find the number of each coin.

Matrix Application Example

Answer 1

N + D+ Q =74.05N +.10D + .25Q = 8.85N- D+ Q = 4

=

=

When a nxn matrix with 1’s on the main diagonal and 0’s everywhere else, it is considered an identity matrix. When you multiply it with another matrix, the answer will come out the same.

Identity Matrix and Inverses

=

To find the inverse of a matrix:

Inverse of a Matrix

[𝑎 𝑏𝑐 𝑑]-1 =

1𝑎𝑑−𝑏𝑐 [ 𝑑 −𝑏

−𝑐 𝑎 ]

[4 72 6]

-1

= 1(4∗6)−(7∗2) [ 6 −7

−2 4 ]

=110 [ 6 −7

−2 4 ]

=[ .6 − .7− .2 .4 ]

Inverse of Matrices Examples

Example 1

[ 8 −5−3 2 ]Find the inverse of :

Example 2

Find the inverse of : [ 5 16−1 −3]

Inverse of Matrices Answers

Answer 1

[ 8 −5−3 2 ]Find the inverse of :

Answer 2

Find the inverse of :

= [2 53 8] =[𝟐 𝟓

𝟑 𝟖 ]

[ 5 16−1 −3]= =