### 7.2 Matrices - Wisconsin Rapids Public Schools

Benn Fox
Hannah Weber
Column
3
2
4
-1
0
8
Row
!
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:
2x3
5
0
4
8
6
5
6
4
9
4
-1
0
8
23
6
7
89
45
35
4
8
41
68
3 Rows
This matrix has 3 rows.
This matrix has 3 columns.
3x3
2 Columns
3 Columns
What is the order of the matrices?
7 Rows
This matrix has 7 rows.
This matrix has 2 columns.
7x2
What is the order of the matrices?
Example 1
5 0
4
9
7
6
5 6
4
2
4
8
Example 2
5
0
3
3
5
6
9
7
6
5 6
4
2
4
8
6 Columns
4
2 Rows
5
0
3
3
1 Row
5
6
6 Columns
5 0
There are 2 rows.
There are 6 columns
2x6
There is 1 rows
There are 6 columns
1x6
a11
a12
a13 . . . . a1n
a21
a22
a23 . . . . a2n
am3
....
am2
....
....
....
am1
amn
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
Because the first letter is 1,
that means it’s in the first
row.
For an example: Identify the element
specified for the following matrix.
a13
Because the second letter is 3,
that means it’s in the third
column.
-2
-6
-4
-5
-2
-6
-4
-5
4
7
9
8
4
7
9
8
9
7
6
5
9
7
6
5
-2
-6
-4
-5
4
7
9
8
9
7
6
5
Because it’s in the 1st row,
and the 3rd column, the
-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,
4
Example 1
Identify the element specified
for the following matrix:
a44
Example 2
8
9
7
-4
Identify the element specified
for the following matrix:
a15
5
8
2
0
43
5
7
-4
8
4
12
3
25
5
4
322
0
3
2
6
8
4
5
45
3
25
6
2
3
8
9
3
543
8
4
55
Identify the element specified
for the following matrix:
a44
8
9
7
-4
Identify the element specified
for the following matrix:
a15
5
8
2
0
43
5
7
-4
8
4
12
3
25
5
4
322
0
3
2
6
8
4
5
45
3
25
6
2
3
8
9
3
543
8
4
55
Because it’s in the 4th row.
Because it’s in the 4th column.
4
Because it’s in the 1st row.
Because it’s in the 4th column.
8
To add or subtract matrices they need to have:
 The same sized rows
 The same sized columns
7 + 6 = 13
7
2
9
4
!
6
1
1
0
13
3
10
4
Add or subtract the numbers in the matching
positions.
2
6
4
9
8
12
5
15
4
6
9
18
2+5 = 7, 4+4=8, 8+9=17
6+15=21, 9+6=15, 12+18=30
9
3
5
4
2
4
0
4
1
9-0=9,
4-7=-3
3-4=-1, 2-1=1
5-1=4,
4-3=1
7
1
3
7
21
8
15
17
30
Example 1
4
7
10
Example 2
5
3
5
8
11
8
12
6
9
12
15
14
3
2
1
6
4
5
1
33
18
13
9
6
10
1
10
4
7
10
5
8
11
3
2
1
6
9
12
6
4
5
9
6
10
4+3=7, 5+6=11, 6+9=15
7+2=9, 8+4=12, 9+6=15
10+1=11, 11+5=16, 12+10=22
5
3
8
12
15
14
1
33
18
13
1
10
5-1=4, 8-18=-10, 15-1=14
3-33=-30, 12-13=-1, 1410=4
4
−30
7
9
11
−10
−1
11
12
16
14
4
15
15
22
!
Columns of the first matrix equals
the number of rows in the second
matrix
4
5
3 4
6 6
2x3
9
2
6
4
3
4
3x2
The same numbers,
mean you can
multiple.
!
The outer numbers
show what dimensions
!
1.
2.
3.
Make sure the number of columns in the 1st
matrix is equal to the number of rows in the
2nd matrix.
Multiply the numbers of each row of the 1st
matrix with the numbers of each column in
the second matrix.
Add up the products from step 2.
1
0
2 3
1 −1
2x3
1 0
2 1
0 −1
3x2
Multiply each number from
the first row in the 1st matrix
with the 1st column of the 2nd
1(1)
1(0)
0(1)
0(0)
+
+
+
+
2(2)
2(1)
1(2)
1(1)
+
+
+
+
3(0) = 5
3(-1) = -1
-1(0) = 2
-1(-1) = 2
Multiply each number from
the first row in the 1st matrix
with the 2nd column of the 2nd
Repeat these steps with the
2nd row of the 1st matrix.
5 −1
2 2
1 0
2 1
0 −1
1(1)
1(2)
1(3)
2(1)
2(2)
2(3)
0(1)
0(2)
0(3)
+
+
+
+
+
+
+
+
+
1
2
0
1
0
2 3
1 −1
0(0) = 1
0(1) = 2
0(-1) = 3
1(0) = 2
1(1) = 5
1(-1) = 5
-1(0) = 0
-1(1) = -1
-1(-1) = 1
2 3
5 5
−1 1
0
−6
2 −2
4 −6
4
5
−2 0
6 −6
0(4) + 2(-2) + -2(6) = -16
0(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
30
2 3
1 −1
You distribute the 3 to
each of the numbers.
3
0
6 9
3 −3
Example 1
−1
3
2
4
Example 2
0 0 1
0 1 0
1 0 0
Example 3
3 4 6
4 4 5 7
5 6 8
−3
5
1 2 1
2 0 1
−1 3 4
−1
3
0 0 1
0 1 0
1 0 0
3
4 4
5
2
4
4 6
5 7
6 8
−3
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
5
1 2 1
2 0 1
−1 3 4
12
16
20
16 24
20 28
24 32
0(1)
0(2)
0(1)
0(1)
0(2)
0(1)
1(1)
1(2)
1(1)
+
+
+
+
+
+
+
+
+
0(2)
0(0)
0(1)
1(2)
1(0)
1(1)
0(2)
0(0)
0(1)
+
+
+
+
+
+
+
+
+
1(-1) = -1
1(3) = 3
1(4) = 4
0(-1) = 2
0(3) = 0
0(4) = 1
0(-1) = 1
0(3) = 2
0(4) = 1
−1 3 4
2 0 1
1 2 1

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.
1st step: assign letters for each variable.
 x=children
 z=senior citizens
 2nd step: set up equations.
.25x+y+.75z=740
x+y+z=1400
x-(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.



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
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.
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.
N + D+ Q =74
.05N +.10D + .25Q = 8.85
N- 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.

To find the inverse of a matrix:

4
2
7
6

-1
-1
1
−

−
−

1
(4 ∗ 6) − (7 ∗ 2)
1
10
6 −7
−2 4
6 −7
−2 4
.6 −.7
−.2 .4
Example 1
Find the inverse of :
8 −5
−3 2
Example 2
Find the inverse of :
5 16
−1 −3
Find the inverse of :
8 −5
−3 2