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We proceed the same way to obtain the second row of A B . In other words, row 2 of A times column 1 of B ; row 2 of A times column 2 of B ; row 2 of A times column 3 of B . When complete, the product matrix will be

A B = [ a 11 b 11 + a 12 b 21 + a 13 b 31 a 21 b 11 + a 22 b 21 + a 23 b 31 a 11 b 12 + a 12 b 22 + a 13 b 32 a 21 b 12 + a 22 b 22 + a 23 b 32 a 11 b 13 + a 12 b 23 + a 13 b 33 a 21 b 13 + a 22 b 23 + a 23 b 33 ]

Properties of matrix multiplication

For the matrices A , B , and C the following properties hold.

  • Matrix multiplication is associative: ( A B ) C = A ( B C ) .
  • Matrix multiplication is distributive: C ( A + B ) = C A + C B , ( A + B ) C = A C + B C .

Note that matrix multiplication is not commutative.

Multiplying two matrices

Multiply matrix A and matrix B .

A = [ 1 2 3 4 ]   and   B = [ 5 6 7 8 ]

First, we check the dimensions of the matrices. Matrix A has dimensions 2 × 2 and matrix B has dimensions 2 × 2. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions 2 × 2.

We perform the operations outlined previously.

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Multiplying two matrices

Given A and B :

  1. Find A B .
  2. Find B A .
A = [ −1 2 3 4 0 5 ]  and   B = [ 5 −4 2 −1 0 3 ]
  1. As the dimensions of A are 2 × 3 and the dimensions of B are 3 × 2 , these matrices can be multiplied together because the number of columns in A matches the number of rows in B . The resulting product will be a 2 × 2 matrix, the number of rows in A by the number of columns in B .
    A B = [ −1 2 3 4 0 5 ]     [ 5 −1 4 0 2 3 ]       = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ]       = [ −7 10 30 11 ]
  2. The dimensions of B are 3 × 2 and the dimensions of A are 2 × 3. The inner dimensions match so the product is defined and will be a 3 × 3 matrix.
    B A = [ 5 −1 −4 0 2 3 ]     [ −1 2 3 4 0 5 ]       = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ]       = [ −9 10 10 4 −8 −12 10 4 21 ]
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Is it possible for AB to be defined but not BA ?

Yes, consider a matrix A with dimension 3 × 4 and matrix B with dimension 4 × 2. For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.

Using matrices in real-world problems

Let’s return to the problem presented at the opening of this section. We have [link] , representing the equipment needs of two soccer teams.

Wildcats Mud Cats
Goals 6 10
Balls 30 24
Jerseys 14 20

We are also given the prices of the equipment, as shown in [link] .

Goal $300
Ball $10
Jersey $30

We will convert the data to matrices. Thus, the equipment need matrix is written as

E = [ 6 30 14 10 24 20 ]

The cost matrix is written as

C = [ 300 10 30 ]

We perform matrix multiplication to obtain costs for the equipment.

C E = [ 300 10 30 ] [ 6 10 30 24 14 20 ]       = [ 300 ( 6 ) + 10 ( 30 ) + 30 ( 14 ) 300 ( 10 ) + 10 ( 24 ) + 30 ( 20 ) ]       = [ 2,520 3,840 ]

The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.

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Given a matrix operation, evaluate using a calculator.

  1. Save each matrix as a matrix variable [ A ] , [ B ] , [ C ] , ...
  2. Enter the operation into the calculator, calling up each matrix variable as needed.
  3. If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.
Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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