# 7.5 Matrices and matrix operations

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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.

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 $\text{\hspace{0.17em}}A,B,\text{}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ are shown below. $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 7\\ 0& -5& 6\\ 7& 8& 2\end{array}\right],C=\left[\begin{array}{c}-1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}3\\ 2\\ 1\end{array}\right]$ ## Describing matrices A matrix is often referred to by its size or dimensions: indicating $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ rows and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ identified as $\text{\hspace{0.17em}}{a}_{ij},\text{}$ we look for the entry in row $\text{\hspace{0.17em}}i,\text{}$ column $\text{\hspace{0.17em}}j.\text{\hspace{0.17em}}$ In matrix $\text{\hspace{0.17em}}A\text{, \hspace{0.17em}}$ shown below, the entry in row 2, column 3 is $\text{\hspace{0.17em}}{a}_{23}.$ $A=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]$ A square matrix is a matrix with dimensions meaning that it has the same number of rows as columns. The $\text{\hspace{0.17em}}3×3\text{\hspace{0.17em}}$ matrix above is an example of a square matrix. A row matrix is a matrix consisting of one row with dimensions $\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]$ A column matrix is a matrix consisting of one column with dimensions $\left[\begin{array}{c}{a}_{11}\\ {a}_{21}\\ {a}_{31}\end{array}\right]$ 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: $\text{\hspace{0.17em}}A,B,C,\text{}$ and so on. Each entry in a matrix is referred to as $\text{\hspace{0.17em}}{a}_{ij},$ such that $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ represents the row and $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}$ represents the column. Matrices are often referred to by their dimensions: $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}×\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ indicating $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ rows and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ columns. ## Finding the dimensions of the given matrix and locating entries Given matrix $\text{\hspace{0.17em}}A:$ 1. What are the dimensions of matrix $\text{\hspace{0.17em}}A?$ 2. What are the entries at $\text{\hspace{0.17em}}{a}_{31}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}_{22}?$ $A=\left[\begin{array}{rrrr}\hfill 2& \hfill & \hfill 1& \hfill 0\\ \hfill 2& \hfill & \hfill 4& \hfill 7\\ \hfill 3& \hfill & \hfill 1& \hfill -2\end{array}\right]$ 1. The dimensions are because there are three rows and three columns. 2. Entry $\text{\hspace{0.17em}}{a}_{31}\text{\hspace{0.17em}}$ is the number at row 3, column 1, which is 3. The entry $\text{\hspace{0.17em}}{a}_{22}\text{\hspace{0.17em}}$ is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column. #### Questions & Answers An investment account was opened with an initial deposit of9,600 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
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
how do they get the third part x = (32)5/4
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
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
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.
a perfect square v²+2v+_
kkk nice
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×