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# Matrices to Represent Data

## Introduction to matrices and related vocabulary

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Defining and Comparing Matrices

For a matinee movie, a movie theater charges the following prices:

Kids: $5 Adults:$8 Seniors: $6 For the same movie at night, the theater charges the following prices: Kids:$7 Adults: $10 Seniors:$8

How could we organize this data to easily compare the prices?

### Matrices

A matrix consists of data that is organized into rows and columns to form a rectangle. For example, we could organize the data collected at a movie theater concession stand during a matinee show into the follow matrix:

\begin{align*}& \quad S \quad M \quad L\\ \begin{matrix} \text{popcorn}\\ \quad \ \text{soda}\end{matrix} & \begin{bmatrix} 20 & 46 & 32\\ 15 & 53 & 29\end{bmatrix}\end{align*}

Now we can easily compare the quantities of each size sold. These values in the matrix are called elements.

This particular matrix has two rows and three columns. Matrices are often described in terms of dimensions (rows by columns). This matrix is a \begin{align*}2 \times 3\end{align*} (read as '2 by 3') matrix.

The variables \begin{align*}m\end{align*} (rows) and \begin{align*}n\end{align*} (columns) are most often used to represent unknown dimensions. Matrices in which the number of rows equals the number of columns \begin{align*}(m=n)\end{align*} are called square matrices.

Matrices which have the same dimensions and all corresponding elements equal are said to be equal matrices.

Let's solve the following problems about matrices.

1. Using the matrix above, what is the value of the element in the second row, second column?

\begin{align*}& \qquad \qquad \qquad \quad {\color{red}{\text{Column 2}}}\\ & \qquad \qquad \quad \quad \qquad \ \ \downarrow\\ & \quad \qquad \qquad \ \ \ \quad S \quad \ \ M \ \ \quad L\\ & \begin{matrix} \qquad \ \ \text{popcorn}\\ {\color{red}{\text{Row 2}}} \rightarrow \text{soda}\end{matrix} \begin{bmatrix} 20 & 46 & 32\\ 15 & \boxed{53} & 29 \end{bmatrix}\end{align*}

We must see where the second row and second column overlap and identify the element in that location. In this case. it is 53.

1. Determine the dimensions \begin{align*}(m \times n)\end{align*} of the matrices below.

1. \begin{align*}\begin{bmatrix} 3 & 2\\ -1 & 0 \end{bmatrix}\end{align*}

This matrix has 2 rows and 2 columns. Therefore it is a \begin{align*}2 \times 2\end{align*} matrix.

1. \begin{align*}\begin{bmatrix} 4 & -3 & 2 & 7\\ 3 & 5 & -4 & 6\\ 9 & 1 & 0 & -2 \end{bmatrix}\end{align*}

This matrix has 3 rows and 4 columns. Therefore it is a \begin{align*}3 \times 4\end{align*} matrix.

1. \begin{align*}\begin{bmatrix} 2\\ -3\\ 1 \end{bmatrix}\end{align*}

This matrix has 3 rows and 1 column. Therefore it is a \begin{align*}3 \times 1\end{align*} matrix.

\begin{align*}A = \begin{bmatrix} 1 & -5\\ -2 & 4\\ 8 & 3 \end{bmatrix} \qquad B = \begin{bmatrix} -5 & 4 & 3\\ 1 & -2 & 8 \end{bmatrix} \qquad C = \begin{bmatrix} 1 & -5\\ -2 & 4\\ 8 & 3 \end{bmatrix}\end{align*}

Matrices \begin{align*}A\end{align*} and \begin{align*}C\end{align*} are equal matrices. They are both \begin{align*}3 \times 2\end{align*} matrices and have all of the same elements. Matrix \begin{align*}B\end{align*} is a \begin{align*}2 \times 3\end{align*} matrix so even though it contains the same elements, they are arranged differently preventing it from being equal to the other two.

### Examples

#### Example 1

Earlier, you were asked how to organize the data to easily compare the prices of a movie theater.

To make it easy to compare prices, we could organize the data in matrix like this one:

\begin{align*} & \ \ \ K \quad A \quad S\\ \begin{matrix} \text{Matinee}\\ \ \ \ \text{Night}\end{matrix} & \begin{bmatrix} 5 & 8 & 6\\ 7 & 10 & 8\end{bmatrix}\end{align*}

#### Example 2

What are the dimensions of the matrix: \begin{align*}[ 3 \quad -5 \quad 1 \quad 0]\end{align*}?

The dimensions are \begin{align*}1 \times 4\end{align*}.

#### Example 3

In the matrix\begin{align*}\begin{bmatrix} 8 & -5 & 4\\ -2 & 6 & -3\\ 3 & 0 & -7\\ 1 & 3 & 9 \end{bmatrix}\end{align*} what is the element in the second row, third column?

The element in the second row, third column is -3 as shown below:

\begin{align*}& \qquad \qquad \qquad \qquad {\color{red}{\text{Column 3}}}\\ & \qquad \quad \qquad \qquad \qquad \ \downarrow\\ & \begin{matrix} {\color{red}{\text{Row 2 }}} \rightarrow \end{matrix} \begin{bmatrix} \\ 8 & -5 & 4\\ -2 & 6 & \boxed{-3}\\ 3 & 0 & -7\\ 1 & 3 & 9\end{bmatrix}\end{align*}

#### Example 4

Are the matrices \begin{align*}A = [-1 \quad 4 \quad 9]\end{align*} and \begin{align*}B = \begin{bmatrix} -1\\ 4\\ 9 \end{bmatrix}\end{align*} equal matrices?

No, \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are not equal matrices. They have the same elements, but the dimensions are not the same.

### Review

Use the matrices below to answer questions 1-7 that follow:

\begin{align*}A = \begin{bmatrix} 2 & 3 & 1\\ -5 & -8 & 4 \end{bmatrix} \qquad B = \begin{bmatrix} 2 & 1\\ -3 & 5 \end{bmatrix} \qquad C = \begin{bmatrix} -5 & 1 & 3\\ 8 & -2 & 6\\ 4 & 9 & 7 \end{bmatrix}\end{align*}

\begin{align*}D = \begin{bmatrix} 2 & 1\\ -3 & 5 \end{bmatrix} \qquad E = \begin{bmatrix} -5 & 2\\ -8 & 3\\ 4 & 1 \end{bmatrix} \qquad F = \begin{bmatrix} 5 & -1 & 8\\ -2 & 6 & -3\\ \end{bmatrix}\end{align*}

1. What are the dimensions of
1. Matrix \begin{align*}B\end{align*}?
2. Matrix \begin{align*}E\end{align*}?
3. Matrix \begin{align*}F\end{align*}?
2. Which matrices have the same dimensions?
3. Which matrices are square matrices?
4. Which matrices are equal?
5. What is the element in row 1, column 2 of Matrix \begin{align*}C\end{align*}?
6. What is the element in row 3, column 1 of Matrix \begin{align*}E\end{align*}?
7. What is the element in row 1, column 1 of Matrix \begin{align*}D\end{align*}?
8. Write a matrix with dimensions \begin{align*}3 \times 4\end{align*}.
9. Write a matrix with dimensions \begin{align*}7 \times 2\end{align*}.

For problems 10-14, determine if the statements are true or false.

1. A \begin{align*}3 \times 2\end{align*} and a \begin{align*}2 \times 3\end{align*} are equal.
2. Two matrices are equal if every element within the two matrices is the same.
3. A matrix is a way to organize data.
4. The element in row 2, column 2 in \begin{align*}F\end{align*} above is -1.
5. The element in row 2, column 2 in \begin{align*}F\end{align*} above is 6.
6. Organize the data into a matrix: A math teacher gave her class three tests during the semester. On the first test there were 10 A’s, 8 B’s, 12 C’s, 4 D’s and 1 F. On the second test there were 8 A’s, 11 B’s, 14 C’s, 2 D’s and 0 F’s. On the third test there were 13 A’s, 7 B’s, 8 C’s, 4 D’s and 3 F’s.

To see the Review answers, open this PDF file and look for section 4.1.

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### Vocabulary Language: English

TermDefinition
diagonal matrix A diagonal matrix is a matrix composed of zeroes in all positions aside from along the diagonal.
Dimensions The number of rows, $m$, and columns, $n$, in a matrix.
Elements The values (numbers) in a matrix.
Equal matrices Equal matrices are matrices that have the same dimensions and elements.
identity matrix An identity matrix is a matrix with zeros everywhere except along the diagonal where there are ones.
Matrices Matrices are rectangular arrangements of data elements presented in rows and columns.
Matrix A matrix is a rectangular arrangement of data elements presented in rows and columns.
order The order of a matrix describes the number of rows and the number of columns in the matrix.
Square matrices Square matrices are matrices in which the number of rows equals the number of columns.
Square matrix A square matrix is a matrix in which the number of rows equals the number of columns.
symmetric matrix A symmetric matrix is a square matrix with reflection symmetry across the main diagonal.
transpose of a matrix The transpose of a matrix is a new matrix whose columns and rows have been switched. This changes the order of the matrix from, for example, $3 \times 2$ to $2 \times 3$.
triangular matrix A triangular matrix is described as either upper or lower triangular . The opposite portion of the matrix is entirely zeros.