# 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 Need help solving this problem (2/7)^-2 Simone Reply what is the coefficient of -4× Mehri Reply -1 Shedrak the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1 Alfred Reply 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?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
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
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
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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opoku
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Ali
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Ali
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Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_