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In this section, you will:
  • Find the sum and difference of two matrices.
  • Find scalar multiples of a matrix.
  • Find the product of two matrices.
(credit: “SD Dirk,” Flickr)

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. [link] shows the needs of both teams.

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

A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the sum and difference of two matrices

To solve a problem like the one described for the soccer teams, we can use a matrix    , which is a rectangular array of numbers. A row    in a matrix is a set of numbers that are aligned horizontally. A column    in a matrix is a set of numbers that are aligned vertically. Each number is an entry    , sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named A , B , and C are shown below.

A = [ 1 2 3 4 ] , B = [ 1 2 7 0 −5 6 7 8 2 ] , C = [ −1 0 3 3 2 1 ]

Describing matrices

A matrix is often referred to by its size or dimensions:   m   ×   n   indicating m rows and n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix A identified as a i j , we look for the entry in row i , column j . In matrix A ,   shown below, the entry in row 2, column 3 is a 23 .

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

A square matrix is a matrix with dimensions   n   ×   n , meaning that it has the same number of rows as columns. The 3 × 3 matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions 1   ×   n .

[ a 11 a 12 a 13 ]

A column matrix is a matrix consisting of one column with dimensions m   ×   1.

[ a 11 a 21 a 31 ]

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

Matrices

A matrix    is a rectangular array of numbers that is usually named by a capital letter: A , B , C , and so on. Each entry in a matrix is referred to as a i j , such that i represents the row and j represents the column. Matrices are often referred to by their dimensions: m × n indicating m rows and n columns.

Finding the dimensions of the given matrix and locating entries

Given matrix A :

  1. What are the dimensions of matrix A ?
  2. What are the entries at a 31 and a 22 ?
    A = [ 2 1 0 2 4 7 3 1 2 ]
  1. The dimensions are   3   ×   3   because there are three rows and three columns.
  2. Entry a 31 is the number at row 3, column 1, which is 3. The entry a 22 is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.
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Questions & Answers

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?
Shirleen Reply
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
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
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
hi
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
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon 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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