Chapter 2 Matrices
2.3 Matrix Operations
Learning Objectives
By the end of this section, you will be able to:
- Identify the dimensions of a matrix
- Identify the entries of a matrix
- Add and subtract matrices
- Calculate a scalar multiple of a matrix
- Multiply matrices
Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. The table below shows the needs of both teams.
[latex]\begin{array}{lcc} &\text{Wildcats}&\text{Mud Cats}\\ \text{Goals}&6&10\\ \text{Balls} &30&24\\ \text{Jerseys}&14&20 \end{array}[/latex]
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 the last section, we learned how matrices can be used to solve systems of linear equations. In this section, we will explore other uses of matrices and 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 in a way that can be easily translated to a computer or calculator.
To solve a problem like the one in the section opener, 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 [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] are shown below.
[latex]A=\begin{bmatrix}1&2\\3&4\end{bmatrix}[/latex] [latex]B=\begin {bmatrix}1&2&7\\0&-5&6\\7&8&2 \end{bmatrix}[/latex] [latex]C=\begin{bmatrix}-1&3\\0&2\\3&1 \end{bmatrix}[/latex]
Describing Matrices
A matrix is often referred to by its size or dimensions: [latex]m\times n[/latex] indicating [latex]m[/latex] rows and [latex]n[/latex] columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix [latex]A[/latex] identified as [latex]a_{ij}[/latex], we look for the entry in row [latex]i[/latex] column [latex]j[/latex]. In matrix [latex]A[/latex], shown below, the entry in row 2, column 3 is [latex]a_{23}[/latex].
[latex]\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{bmatrix}[/latex]
A square matrix is a matrix with dimensions [latex]n \times n[/latex], meaning that it has the same number of rows as columns. The matrix [latex]3 \times 3[/latex] above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions [latex]1 \times n[/latex].
[latex]\begin {bmatrix} a_{11}&a_{12}&a_{13} \end{bmatrix}[/latex]
A column matrix is a matrix consisting of one column with dimensions [latex]m \times 1[/latex].
[latex]\begin{bmatrix} a_{11}\\a_{21}\\a_{31} \end{bmatrix}[/latex]
A matrix may be used to represent a system of equati