Matrix Multiplication

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Matrix Multiplication
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Matrix multiplication involves both
multiplication and addition. The process can be
demonstrated using football scores. Suppose a
team scores 5 touchdowns, 4 extra points, and 2
field goals. To find the total points scored
multiply
(5)(6) (4)(1) (2)(3) 30 4 6 40 since
TDs are worth 6 points, extra points are worth 1
point, and field goals are worth 3 points.
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To model this with matrix multiplication,
represent the number of TDs, extra points, and
field goals in a 1 x 3 matrix. Represent the
point values of each in a 3 x 1 matrix.
Notice that the product of a 1 x 3 matrix and a 3
x 1 matrix is a 1 x 1 matrix.
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To determine if it is possible to multiply
matrices it is necessary to look at their
dimensions.
Matrices may be multiplied only if the inner
dimensions are the same.
Outer dimensions 1 x 3 3 x 1 Inner
dimensions
The dimensions of the product matrix are the same
as the outer dimensions of the factors.
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Is it possible to find the product AB?
Write the product with the matrix dimensions
This product is not possible because the inner
dimensions are different.
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Is it possible to find the product BA?
Write the product with the matrix dimensions
This product is possible because the inner
dimensions are the same.
The product will be a 2 x 3 matrix.
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Matrix Multiplication If matrix A has dimensions
m x n and matrix B has dimensions n x r, then the
product AB has dimensions m x r.
Find the entry in row i and column j of AB by
finding the sum of the products of the
corresponding entries in row i of A and column j
of B.
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Find HG, if it exists.
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Write the product matrix with the row/column
designation of each entry.
Notice that this is the product of row 1 and
column 1.
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Notice that this is the product of row 1 and
column 2.
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Notice that this is the product of row 2 and
column 1.
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Notice that this is the product of row 2 and
column 2.
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Therefore, the product HG
Is the product GH the same thing?
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Find GH, if it exists.
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Write the product matrix with the row/column
designation of each entry.
Notice that this is the product of row 1 and
column 1.
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Notice that this is the product of row 1 and
column 2.
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Notice that this is the product of row 2 and
column 1.
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Notice that this is the product of row 2 and
column 2.
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Therefore, the product GH
Is matrix multiplication commutative?
Even though both HG and GH exist they are not
equal. Therefore, matrix multiplication is not
commutative.
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Find RW, if it exists.
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Find WR, if it exists.
WR does not exist.
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Matrix Multiplication - Application
Karl and Kayla are making two snack mixes by
mixing dried fruit and nuts. The amounts of
protein, carbohydrates, and fat, in grams per
serving, for the dried fruit and nuts are given
in Table 1. The number of servings of dried
fruit and nuts in each is given in Table 2.
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Table 1
Dried Fruit Nuts
Protein 3 20
Carbohydrates 65 21
Fat 1 52
Table 2
Sports Mix Camp Mix
Dried Fruit 4 3
Nuts 2 3
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Represent the information from Table 1 in a
matrix called N. Represent the information from
Table 2 in a matrix called G.
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Find the product NG, and determine which mix has
more protein and which mix has less fat.
The camp mix has more protein, 69 grams. The
sport mix has less fat, 108 grams.
Sport Mix
Camp Mix
Protein Carbohydrate Fat