# 7.5 Matrices and matrix operations  (Page 4/10)

 Page 4 / 10

We proceed the same way to obtain the second row of $\text{\hspace{0.17em}}AB.\text{\hspace{0.17em}}$ In other words, row 2 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ times column 1 of $\text{\hspace{0.17em}}B;\text{\hspace{0.17em}}$ row 2 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ times column 2 of $\text{\hspace{0.17em}}B;\text{\hspace{0.17em}}$ row 2 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ times column 3 of $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ When complete, the product matrix will be

$AB=\left[\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}\\ \end{array}\\ {a}_{21}\cdot {b}_{11}+{a}_{22}\cdot {b}_{21}+{a}_{23}\cdot {b}_{31}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}\\ \end{array}\\ {a}_{21}\cdot {b}_{12}+{a}_{22}\cdot {b}_{22}+{a}_{23}\cdot {b}_{32}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}\\ \end{array}\\ {a}_{21}\cdot {b}_{13}+{a}_{22}\cdot {b}_{23}+{a}_{23}\cdot {b}_{33}\end{array}\right]$

## Properties of matrix multiplication

For the matrices $\text{\hspace{0.17em}}A,B,\text{}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ the following properties hold.

• Matrix multiplication is associative: $\text{\hspace{0.17em}}\left(AB\right)C=A\left(BC\right).$
• Matrix multiplication is distributive: $\begin{array}{l}\begin{array}{l}\\ \text{\hspace{0.17em}}C\left(A+B\right)=CA+CB,\end{array}\hfill \\ \text{\hspace{0.17em}}\left(A+B\right)C=AC+BC.\hfill \end{array}$

Note that matrix multiplication is not commutative.

## Multiplying two matrices

Multiply matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and matrix $\text{\hspace{0.17em}}B.$

First, we check the dimensions of the matrices. Matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ has dimensions $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ and matrix $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ has dimensions $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2.$

We perform the operations outlined previously.

## Multiplying two matrices

Given $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B:$

1. Find $\text{\hspace{0.17em}}AB.$
2. Find $\text{\hspace{0.17em}}BA.$
1. As the dimensions of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}2\text{}×\text{}3\text{\hspace{0.17em}}$ and the dimensions of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}3\text{}×\text{}2,\text{}$ these matrices can be multiplied together because the number of columns in $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ matches the number of rows in $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ The resulting product will be a $\text{\hspace{0.17em}}2\text{}×\text{}2\text{\hspace{0.17em}}$ matrix, the number of rows in $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the number of columns in $\text{\hspace{0.17em}}B.$
2. The dimensions of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}3×2\text{\hspace{0.17em}}$ and the dimensions of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}2×3.\text{\hspace{0.17em}}$ The inner dimensions match so the product is defined and will be a $\text{\hspace{0.17em}}3×3\text{\hspace{0.17em}}$ matrix.

Is it possible for AB to be defined but not BA ?

Yes, consider a matrix A with dimension $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ and matrix B with dimension $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ 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 $E=\left[\begin{array}{c}6\\ 30\\ 14\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}10\\ 24\\ 20\end{array}\right]$ The cost matrix is written as $C=\left[\begin{array}{ccc}300& 10& 30\end{array}\right]$ We perform matrix multiplication to obtain costs for the equipment. The total cost for equipment for the Wildcats is$2,520, and the total cost for equipment for the Mud Cats is $3,840. Given a matrix operation, evaluate using a calculator. 1. Save each matrix as a matrix variable $\text{\hspace{0.17em}}\left[A\right],\left[B\right],\left[C\right],...$ 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?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
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Carole
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I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
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sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
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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
oops. ignore that.
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Abhi
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Abhi
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Abhi
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salma
Commplementary angles
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Sherica
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Tamia
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Uday
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salma
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×