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4.4 Multiplication of Matrices

Algebra 2

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Learning Target

• I can multiply two matrices and interpret the results.

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Application• Mac MacDonough owns three fruit farms on which

he grows peaches, apricots, plums and apples. When picked, the fruit is sorted into layered box in which they will be sold. The chart below shows the number of boxes for each fruit type.

Location Peaches Apricots Plums Apples

Farm 1 165 217 430 290

Farm 2 243 190 235 175

Farm 3 74 150 198 0

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Application

• Suppose he sells peaches for $26 a box, apricots for $18 per box, plums for $32 a box and apples for $19 a box. The total income from this picking of fruit could be found by multiplying matrices.

Location Peaches Apricots Plums Apples

Farm 1 165 217 430 290

Farm 2 243 190 235 175

Farm 3 74 150 198 0

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Some tips

• Writing the first matrix order as m x n and the second matrix order as n x r may help you to see that the result is

m x r.• You can multiply two matrices only if

the number of columns in the first matrix is equal to the number of rows in the second matrix.

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Multiplying Matrices (The rule)

• The product of a m x n matrix, A and an n x r matrix, B, is the m x r matrix AB. The element in the ith row and the fth column of AB is the sum of the products of the corresponding elements in the ith row of A and the ith row column of B. Multiplication of matrices IS NOT COMMUTATIVE.

• The steps in example 1 will illustrate how two matrices can be multiplied.

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Ex. 1: If and , find AB.

4312

A

675293

B

323222 )(1812910251

24628272096471856

)6(4)2(3)7(4)9(3)5(4)3(3)6)(1()2(2)7)(1()9(2)5)(1()3(2

xxx ABBA

AB

AB

AB

Matrix BA is not defined since B has 3 columns and A has 2 rows.

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Ex. 2: If and , find AB.

5321086

A

263352

B

A has 3 columns and B has 2 rows. In order to find AB, A must have the same number of columns as B has rows. Since this is not the case, AB is not defined.

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Ex. 3: Find the total income of the three fruit farms owned by Mr. McDonough.

• The first matrix represents the numbers of boxes of each type fruit for each farm. The second matrix will list the prices per box for each type of fruit.

960,10583,20466,27

19321826

019815074175235190243290430217165

A3x4 B4x1 (AB)3x1

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Ex. 3: Find the total income of the three fruit farms owned by Mr. McDonough.

• Farm 1 earned $27,466. Farm 2 earned $20,583, and Farm 3 earned $10,960. The total income is $59,009.

960,10583,20466,27

19321826

019815074175235190243290430217165

A3x4 B4x1 (AB)3x1

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Other uses of matrix multiplication

• Another use of matrix multiplication is in transformational geometry. You have already learned to translate a figure and change the size of a figure using matrices. When you wish to move a figure by rotating it, you can use a rotation matrix.

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The matrix will rotate a figure.

• On a coordinate plane 90º counterclockwise, about the origin. In the figure to your right, segment AB is rotate 90º counterclockwise, using the origin as the point of rotation. The result is segment A’B’.

• Segments AO and OA’ form a 90º angle. Likewise, segments OB and OB’ form a 90º angle.

8

6

4

2

-2

-4

-6

-8

-5 5 10

0110

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Ex. 4: Triangle RST has vertices with coordinates R(-1, -2), S(2, -4), and T(5, 3).

• Find the coordinates of the vertices of this triangle after it is rotated counterclockwise 90º about the origin.

4

2

-2

-4

-6

-8

-10

-12

5 10 15

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Ex. 4: Triangle RST has vertices with coordinates R(-1, -2), S(2, -4), and T(5, 3).

• Let each column of a matrix represent an ordered pair of the triangle with the top row containing the x-values. Then multiply the coordinate matrix by the rotation matrix.

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0(-1)+(-1)(-2) = 0 + 2 = 2

0(2)+(-1)(-4) = 0 + 4 = 4

0(5)+(-1)(3) = 0 - 3 = -31(-1)+(0)(-2) = -1 + 0 = -1

1(2)+(0)(-4) = 2 + 0 = 2

1(5)+(0)(3) = 5 + 0 = 5The coordinates of the vertices of the rotated triangle are R’(2, 01), S’(4, 2), and T’(-3, 5).

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The rotation

4

2

-2

-4

-5 5

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Ex. 5: Triangle ABC has vertices with coordinates A(3, 4), B(6, 5), and C(0, 0).

• Let each column of a matrix represent an ordered pair of the triangle with the top row containing the x-values. Then multiply the coordinate matrix by the rotation matrix.

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054063

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0(3)+(-1)(4) = 0 - 4 = -4

0(6)+(-1)(5) = 0 - 5 = -5

0(0)+(-1)(0) = 0 - 0 = 01(3)+(0)(4) = 3 + 0 = 3

1(6)+(0)(5) = 6 + 0 = 6

1(0)+(0)(0) = 0 + 0 = 0The coordinates of the vertices of the rotated triangle are A’(-4, 3), B’(-5, 6), and C’(0, 0).

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What if?

• You can’t hack it? Come to tutoring. We will be doing homework at lunch and after school. If you don’t show up until December 15th, then it’s kind of obvious, to me anyway, that you really aren’t that interested in getting your scores any higher.

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Pair-share

• Pg. 176 #13-24 odds

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Resource

• http://www.taosschools.org/ths/Departments/MathDept/spitz/Syllabi/Algebra2_PPTs.htm

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