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# 4.1: Defining and Comparing Matrices

Difficulty Level: At Grade Created by: CK-12
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Practice Matrices to Represent Data

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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?

### Watch This

Watch the first part of this video, until about 4:45.

### Guidance

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 \ }\\ \text{ \quad 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 its 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.

#### Example A

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

Solution:

\begin{align*}& \qquad \qquad \qquad \qquad {\color{red}{\text{Column 2}}}\\ & \qquad \qquad \quad \qquad \qquad \ \ \downarrow\\ & \quad \qquad \qquad \qquad \quad S \quad M \quad L\\ & \begin{matrix} \text{\qquad \ \ 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.

#### Example B

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

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

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

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

Solution:

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

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

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

#### Example C

\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*}

Solution:

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.

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

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

### Guided Practice

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

2. 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?

3. 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?

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

2. 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*}

3. 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.

### Vocabulary

Matrix
A rectangular arrangement of data elements presented in rows and columns.
Elements
The values (numbers) in a matrix.
Dimensions
The number of rows, \begin{align*}m\end{align*}, and columns, \begin{align*}n\end{align*}, in a matrix.
Square matrices
Matrices in which the number of rows equals the number of columns, or \begin{align*}m=n\end{align*}.
Equal matrices
Matrices which have the same dimensions and elements.

### Practice

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*}.

Determine if the following statements are true or false.

1. A \begin{align*}3x2\end{align*} and a \begin{align*}2x3\end{align*} are equal.
2. Two matrices are equal if every element within the two matrices are 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.

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Color Highlighted Text Notes

### Vocabulary Language: English

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.

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