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.

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1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
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