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**.

#### Solve the following problems

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.

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

\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.

\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.

\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.

Which two matrices are equal? Explain your answer.

\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 could we organize the data to easily compare the prices.

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*}

- What are the dimensions of
- Matrix \begin{align*}B\end{align*}?
- Matrix \begin{align*}E\end{align*}?
- Matrix \begin{align*}F\end{align*}?

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

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

- A \begin{align*}3 \times 2\end{align*} and a \begin{align*}2 \times 3\end{align*} are equal.
- Two matrices are equal if every element within the two matrices is the same.
- A matrix is a way to organize data.
- The element in row 2, column 2 in \begin{align*}F\end{align*} above is -1.
- The element in row 2, column 2 in \begin{align*}F\end{align*} above is 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.

### Answers for Review Problems

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