EQ: How do I multiply matrices??
CCGPS PreCalculus
Name _____________________
Date __________________
In order to multiply matrices the number of ______________ in the first matrix must equal the
number of ______________ in the second. The resulting matrix is the number of
______________of the first and the number of ____________ in the second.
Given the matrices with the following dimensions, which ones could be multiplied?
Dimensions of A
Dimensions of B
1) 2 x 3
2x1
2) 3 x 3
3x1
3) 2 x 2
2x3
4) 1 x 2
3x2
5) 2 x 4
4x3
6) 3 x 1
1x2
Can you multiply?
Dimensions of AB
Multiplying Matrices:
Example One:
A=
⎡4 − 5⎤
⎢6 7 ⎥
⎣
⎦
B=
⎡ 2 3⎤
⎢ − 1 4⎥
⎣
⎦
What are the dimensions of A? ____________ Dimensions of B? ________________
Dimensions of AB? _________________
Example Two:
A=
⎡ 2 − 5⎤
⎢− 3 4 ⎥
⎣
⎦
B=
⎡3 4 3⎤
⎢ − 2 1 − 2⎥
⎣
⎦
What are the dimensions of A? _______________
Dimensions of B? _______________
Dimensions of AB? ____________________
Example Three:
⎡ 2 4 − 1⎤
⎢
A= 7
6 4 ⎥⎥
⎢
⎢⎣− 2 1 5 ⎥⎦
⎡0⎤
⎢2⎥
B=
⎢ ⎥
⎢⎣− 8⎥⎦
What are the dimensions of A? _______________
Dimensions of B? _______________
Dimensions of AB? ____________________
In order to multiply matrices, you must multiply each entry of each row of the first matrix by
each entry of each column of the second matrix. Then you sum each product and place in the
appropriate entry of the resulting matrix.
EQ: How do I multiply matrices??
CCGPS PreCalculus
Name _____________________
Date __________________
Example One:
A=
Find AB.
⎡4 − 5⎤
⎢6 7 ⎥
⎣
⎦
B=
⎡ 2 3⎤
⎢ − 1 4⎥
⎣
⎦
⎡ _____ ____ ⎤
AB
AB
⎢ _____
____ ⎥⎦
⎣ AB
AB
11
12
21
22
EQ: How do I multiply matrices??
CCGPS PreCalculus
Name _____________________
Date __________________
Example 2:
A=
⎡ 2 − 5⎤
⎢− 3 4 ⎥
⎣
⎦
B=
⎡3 4 3⎤
⎢ − 2 1 − 2⎥
⎣
⎦
Find AB.
⎡ _____ ____ ____ ⎤
AB ⎥
AB
AB
⎢ _____
____ ____ ⎦
⎣ AB
AB
AB
11
12
13
21
22
23
You Do Example 3 Below:
⎡ 2 4 − 1⎤
⎢
A= 7
6 4 ⎥⎥
⎢
⎢⎣− 2 1 5 ⎥⎦
Find AB.
⎡0⎤
⎢2⎥
B=
⎢ ⎥
⎢⎣− 8⎥⎦
EQ: How do I multiply matrices??
CCGPS PreCalculus
EOCT Problems:
Name _____________________
Date __________________
Find the missing entries:
⎡3 5 ⎤
1) A = ⎢
⎥
⎣2 x ⎦
⎡ 2 3 − 5⎤
2) A = ⎢− 1 0 x ⎥
⎢
⎥
⎢⎣ 2 y 6 ⎥⎦
3) A =
[.5
⎡22 − 20 ⎤
⎢10 − 11 ⎥
⎣
⎦
⎡4 y ⎤
B= ⎢
⎥ AB =
⎣2 −1⎦
3 − 4]
⎡1⎤
B= ⎢ 2 ⎥
⎢ ⎥
⎢⎣− 5⎥⎦
⎡ 33 ⎤
AB = ⎢− 6⎥
⎢ ⎥
⎢⎣ 0 ⎥⎦
⎡ a 6 − 2 8⎤
B = ⎢ 4 − 3 b 8⎥
⎢
⎥
⎢⎣7 0
1 c ⎥⎦
AB = [− 15 − 6 22 44]
4) Matrix T shows the number of employees (in thousands) in several areas of the transportation industry (by
columns: air, water, truck, and ground passenger) for 1998 and 1999 (rows). Matrix S shows the average
annual salary for employees in each transportation area (by rows: air, water, truck, and ground passenger.
a.
Which entry of Matrix TS will show how much the transportation industry spent on salaries in
1998?
b.
Which entry of Matrix TS will show how much the transportation industry spend on salaries in
1999?
Air
Water Truck Ground
Salaries
⎡75,000⎤
⎢53,000⎥
Water
⎢
⎥
S=
⎢98,000⎥
Truck
⎢
⎥
Ground ⎣62,000⎦
Air
T=
1998
1999
⎡150 136 398 250⎤
⎢160 148 580 300⎥
⎣
⎦