# Slide 1 - The Pedaled Boxcar Challenge by yurtgc548

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```									EXAMPLE 1     Add and subtract matrices

Perform the indicated operation, if possible.
3 0     –1 4     3 + (– 1)   0+4       2      4
a. – 5 – 1 + 2 0 =      –5+2 –1+0 = –3            –1

7 4  –2    5       7 – (– 2)          4–5        9 –1
b. 0 –2 – 3 – 10 =         0–3      – 2 – (– 10) = – 3 8
–1 6  –3    1      – 1 – (– 3)          6–1       2 5
EXAMPLE 2     Multiply a matrix by a scalar

Perform the indicated operation, if possible.
4–1     – 2(4) – 2(– 1) –8      2
a. – 2 1 0 = – 2(1) – 2(0) = – 2       0
2 7     – 2(2) – 2(7)   – 4 – 14

–2 –8   –3
+ –3 8
8          4(– 2) 4(– 8)
b. 4       +            =
5 0     6 –5            4(5)   4(0)    6 –5

– 8 – 32   –3       8
=          +
20     0    6      –5
EXAMPLE 2   Multiply a matrix by a scalar

– 8 + (– 3)      – 32 + 8
=
20 + 6       0 + (– 5)

– 11 – 24
=
26   –5
EXAMPLE 3     Solve a multi-step problem

Manufacturing
A company manufactures small and large steel DVD
racks with wooden bases. Each size of rack is
available in three types of wood: walnut, pine, and
cherry. Sales of the racks for last month and this
month are shown below.
EXAMPLE 3     Solve a multi-step problem

Organize the data using two matrices, one for last
month’s sales and one for this month’s sales. Then
write and interpret a matrix giving the average
monthly sales for the two month period.
SOLUTION
STEP 1 Organize the data using two 3 X 2 matrices, as
shown.
Last Month (A)        This Month (B)
Small      Large        Small     Large
Walnut 125          100            95      114
Pine     278        251           316      215
Cherry 225          270           205      300
EXAMPLE 3     Solve a multi-step problem

STEP 2 Write a matrix for the average monthly sales
by first adding A and B to find the total sales
and then multiplying the result by 1 .
2
125 100       95       114
1 (A + B) = 1   278 251    + 316       215
2           2   225 270      205       300

220 214
= 1    594 466
2
430 570
EXAMPLE 3     Solve a multi-step problem

110 107
= 297 233
215 285

STEP 3 Interpret the matrix from Step 2. The
company sold an average of 110 small walnut
racks, 107 large walnut racks, 297 small pine
racks, 233 large pine racks, 215 small cherry
racks, and 285 large cherry racks.
EXAMPLE 4      Solve a matrix equation

Solve the matrix equation for x and y.
5x – 2      3   7     – 21    15
3          +            =
6 –4      –5 –y          3 – 24

SOLUTION

Simplify the left side of the equation.

5x   –2    3    7       – 21   15     Write original
3           +           =                 equation.
6   –4   –5   –y          3 – 24
EXAMPLE 4        Solve a matrix equation

5x + 3          5         – 21   15 Add matrices inside
3                         =
1     –4 –y            3 – 24 parentheses.

15x + 9          15       – 21   15 Perform scalar
=
3        – 12 – 3y          3 – 24 multiplication.

Equate corresponding elements and solve the two
resulting equations.
15x + 9 = – 21          – 12 – 3y = -24
x=–2                      y=4

The solution is x = – 2 and y = 4.
EXAMPLE 1     Describe matrix products

State whether the product AB is defined. If so, give
the dimensions of AB.
a. A: 4 3, B: 3 2                  b. A: 3 4, B: 3 2

SOLUTION
a. Because A is a 4 3 matrix and B is a 3 2 matrix,
the product AB is defined and is a 4 2 matrix.

b. Because the number of columns in A (four) does
not equal the number of rows in B (three), the
product AB is not defined.
EXAMPLE 2     Find the product of two matrices

1 4                5 –7
Find AB if A =           and B =
3 –2               9  6

SOLUTION

Because A is a 2 2 matrix and B is a 2 2 matrix, the
product AB is defined and is a 2 2 matrix.
EXAMPLE 2      Find the product of two matrices

STEP 1

Multiply the numbers in the first row of A by the
numbers in the first column of B, add the products,
and put the result in the first row, first column of AB.

1    4     5 –7         1(5) + 4(9)
=
3   –2     9 6
EXAMPLE 2     Find the product of two matrices

STEP 2

Multiply the numbers in the first row of A by the
numbers in the first column of B, add the products,
and put the result in the first row, second column of
AB.

1    4   5 –7      1(5) + 4(9)     1( – 7) + 4(6)
=
3   –2   9 6
EXAMPLE 2      Find the product of two matrices

STEP 3

Multiply the numbers in the second row of A by the
numbers in the first column of B, add the products,
and put the result in the second row, first column of
AB.

1    4   5 –7     1(5) + 4(9)       1( – 7) + 4(6)
=
3   –2   9 6      3(5) + (– 2)(9)
EXAMPLE 2       Find the product of two matrices

STEP 4

Multiply the numbers in the second row of A by the
numbers in the second column of B, add the
products, and put the result in the second row,
second column of AB.

1    4   5 –7    1(5) + 4(9)          1( – 7) + 4(6)
= 3(5) + (– 2)(9)
3   –2   9 6                         3( –7) + ( –2)(6)
EXAMPLE 2       Find the product of two matrices

STEP 5

1(5) + 4(9)         1(– 7) + 4(6)          41 17
=
3(5) + (–2)(9)      3(–7) + (–2)(6)        –3 –33
EXAMPLE 3         Use matrix operations

Using the given matrices, evaluate the expression.

4        3          –3     0        1     4
A= –1       –2 ,B=        1   –2 , C = –3    –1
2        0

a. A(B + C)                                 b. AB + AC

SOLUTION

4     3
a. A(B + C) =    –1    –2       –3       0 + 1     4
2     0        1      –2   –3   –1
EXAMPLE 3       Use matrix operations

4     3        –2    4        –14         7
=   –1    –2        –2   –3   =     6         2
2     0                       –4         8

4       3   –3    0    4        3       1 4
b. AB + AC = –1      –2    1   –2 + –1       –2      –3 –1
2       0              2        0

–9    –6        –5       13        – 14   7
1     4   +     5      –2 =        6     2
–6     0         2       8         –4     8
EXAMPLE 4      Use matrices to calculate total cost

Sports Two hockey
teams submit equipment
lists for the season as
shown.

Each stick costs \$60, each
puck costs \$2, and each
uniform costs \$35. Use matrix
multiplication to find the total
cost of equipment for each
team.
EXAMPLE 4     Use matrices to calculate total cost

SOLUTION

To begin, write the equipment lists and the costs
per item in matrix form. In order to use matrix
multiplication, set up the matrices so that the
columns of the equipment matrix match the rows
of the cost matrix.
EXAMPLE 4    Use matrices to calculate total cost

Equipment
Sticks Pucks Uniforms
Women’s team       14     30    18
Men’s team         16     25    20

Cost
Dollars
Sticks           60
Pucks            2
Uniforms         35
EXAMPLE 4     Use matrices to calculate total cost

The total cost of equipment for each team can be
found by multiplying the equipment matrix by the
cost matrix. The equipment matrix is 2 3 and the
cost matrix is 3 1. So, their product is a 2 1
matrix.

14    30      18           60
16    25      20           2
35

= 14(60) + 30(2) + 18(35) = 1530
16(60) + 25(2) + 20(35)   1710
EXAMPLE 4    Use matrices to calculate total cost

The labels for the product matrix are shown below.

Total Cost
Dollars
Women’s team           1530
Men’s team             1710

ANSWER     The total cost of equipment for the women’s
team is \$1530, and the total cost for the
men’s team is \$1710.

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