MULTIPLYING TWO MATRICES

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MULTIPLYING TWO MATRICES
The product of two matrices A and B is defined
provided the number of columns in A is equal to
the number of rows in B.
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MULTIPLYING TWO MATRICES
A ? B
? AB
4 X 3 ? 3 X 5 ?
4 X 5
4 rows
3 rows
3 columns
5 columns
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MULTIPLYING TWO MATRICES
A ? B
? AB
4 X 3 ? 3 X 5 ?
4 X 5
4 rows
5 columns
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MULTIPLYING TWO MATRICES
If A is a 4 X 3 matrix and B is a 3 X 5 matrix,
then the product AB is a 4 X 5 matrix.
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MULTIPLYING TWO MATRICES
A ? B
? AB
m X n ? n X p ?
m X p
m rows
n rows
n columns
p columns
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MULTIPLYING TWO MATRICES
A ? B
? AB
m X n ? n X p ?
m X p
m rows
p columns
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MULTIPLYING TWO MATRICES
If A is an m X n matrix and B is an n X p
matrix, then the product AB is an m X p matrix.
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Find AB if
A
and B
SOLUTION
Because A is a 3 X 2 matrix and B is a 2 X 2
matrix, the product AB is defined and is a 3 X 2
matrix.
To write the entry in the first row and first
column of AB, multiply corresponding entries in
the first row of A and the first column of B.
Then add.
Use a similar procedure to write the other
entries of the product.
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A ? B ?
AB
3 X 2 ? 2 X 2 ?
3 X 2
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A ? B ?
AB
3 X 2 ? 2 X 2 ?
3 X 2
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A ? B ?
AB
3 X 2 ? 2 X 2 ?
3 X 2
12
A ? B ?
AB
3 X 2 ? 2 X 2 ?
3 X 2
?
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MULTIPLYING TWO MATRICES
Matrix multiplication is not, in general,
commutative.
Let A, B, and C be matrices and let g be a scalar.
ASSOCIATIVE PROPERTY OF MATRIX MULTIPLICATION
A(BC ) (AB )C
A(B C ) AB AC
LEFT DISTRIBUTIVE PROPERTY
(A B )C AC BC
RIGHT DISTRIBUTIVE PROPERTY
ASSOCIATIVE PROPERTY OF SCALAR MULTIPLICATION
g (AB ) (gA )B A(gB )
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USING MATRIX MULTIPLICATION IN REAL LIFE
Matrix multiplication is useful in business
applications because an inventory matrix, when
multiplied by a cost per item matrix, results in
total cost matrix.


m X n
n X p
m X p
For the total cost matrix to be meaningful, the
column labels for the inventory matrix must match
the row labels for the cost per item matrix.
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SPORTS Two softball teams submit equipment lists
for the season.
Each bat costs 21, each ball costs 4, and each
uniform costs 30.
Use matrix multiplication to find the total cost
of equipment for each team.
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SOLUTION
Write the equipment lists and costs per item in
matrix form.Use matrix multiplication to find
the total cost. Set up matrices so that columns
of the equipment matrix match rows of the cost
matrix.
COST
EQUIPMENT
Dollars
Balls
Uniforms
Bats
21
Womens team
12
45
15
Bats
4
15
38
17
Mens team
Balls
30
Uniforms
17
Total equipment cost for each team can be
obtained by multiplying the equipment matrix by
the cost per item matrix. The equipment matrix
is 2 X 3 and the cost per item matrix is 3 X
1. Their product is a 2 X 1 matrix.
12(21) 45(4) 15(30)
882


15(21) 38(4) 17(30)
977
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TOTAL COST
Dollars
The labels of the product are
882
Womens team
977
Mens team
The total cost of equipment for the womens team
is 882,and the total cost of equipment for the
mens team is 977.