2
SYSTEMS OF
LINEAR
EQUATIONS AND
MATRICES
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2.5
Multiplication of Matrices
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Matrix Product
3
Matrix Product
To define matrix multiplication, let’s consider the following
problem.
On a certain day, Al’s Service Station sold 1600 gallons of
regular, 1000 gallons of regular plus, and 800 gallons of
premium gasoline.
If the price of gasoline on this day was $3.09 for regular,
$3.29 for regular plus, and $3.45 for premium gasoline, find
the total revenue realized by Al’s for that day.
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Matrix Product
The day’s sale of gasoline may be represented by the
matrix
A = [1600 1000 800]
Row matrix (1  3)
Next, we let the unit selling price of regular, regular plus,
and premium gasoline be the entries in the matrix
Column matrix (3  1)
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Matrix Product
The first entry in matrix A gives the number of gallons of
regular gasoline sold, and the first entry in matrix B gives
the selling price for each gallon of regular gasoline, so their
product (1600)(3.09) gives the revenue realized from the
sale of regular gasoline for the day.
A similar interpretation of the second and third entries in
the two matrices suggests that we multiply the
corresponding entries to obtain the respective revenues
realized from the sale of regular, regular plus, and premium
gasoline.
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Matrix Product
Finally, the total revenue realized by Al’s from the sale of
gasoline is given by adding these products to obtain
(1600)(3.09) + (1000)(3.29) + (800)(3.45) = 10,994
or $10,994.
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Matrix Product
This example suggests that if we have a row matrix of size
1  n,
A = [a1 a2 a3 … an]
and a column matrix of size n  1,
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Matrix Product
Then we may define the matrix product of A and B, written
AB, by
(12)
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Example 1
Let
A = [1 – 2 3 5] and
Then
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Matrix Product
Returning once again to the matrix product AB in Equation
(12), observe that the number of columns of the row matrix
A is equal to the number of rows of the column matrix B.
Observe further that the product matrix AB has size 1  1
(a real number may be thought of as a 1  1 matrix).
Schematically,
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Matrix Product
More generally, if A is a matrix of size m  n and B is a
matrix of size n  p (the number of columns of A equals the
numbers of rows of B), then the matrix product of A and B,
AB, is defined and is a matrix of size m  p. Schematically,
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Example 3
Let
Compute AB.
Solution:
The size of matrix A is 2  3, and the size of matrix B is
3  3.
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Example 3 – Solution
cont’d
The product AB is given by
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Matrix Product
In general, AB  BA for two square matrices A and B.
However, the following laws are valid for matrix
multiplication.
15
Matrix Product
The square matrix of size n having 1s along the main
diagonal and 0s elsewhere is called the identity matrix of
size n.
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Matrix Product
The identity matrix has the properties that InA = A for every
n  r matrix A and BIn = B for every s  n matrix B.
In particular, if A is a square matrix of size n, then
InA = AIn = A
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Example 5
Let
Then
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Example 5
cont’d
so I3A = AI3 = A, confirming our result for this special case.
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Practice
p. 122 Concept Questions #1 & 2
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