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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.

Questions & Answers

An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
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Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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Abhi
is it a question of log
Abhi
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salma
Commplementary angles
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Uday
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salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
Practice Key Terms 5

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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