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 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.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a [latex]3 \times 3[/latex] matrix and another [latex]3 \times 3[/latex] matrix, but we cannot add or subtract a [latex]2 \times 3[/latex] matrix and a [latex]3 \times 3[/latex] matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Scalar Multiplication

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If [latex]A[/latex] is an [latex]m \times r[/latex] matrix and [latex]B[/latex] is an [latex]r \times n[/latex] matrix, then the product matrix [latex]AB[/latex] is an [latex]m \times n[/latex] matrix. For example, the product [latex]AB[/latex] is possible because the number of columns in [latex]A[/latex] is the same as the number of rows in [latex]B[/latex]. If the inner dimensions do not match, the product is not defined.

[latex]\;\;\;A\;\;\;\; \cdot \;\;\;\; B\;\;\;[/latex]

[latex]2 \times \underbrace {3 \;\;\; \;\;\; 3}_{same}\times 3[/latex]

We multiply entries of [latex]A[/latex] with entries of [latex]B[/latex] according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row [latex]i[/latex] of [latex]AB[/latex], we multiply the entries in row [latex]i[/latex] of [latex]A[/latex] by column [latex]j[/latex] in [latex]B[/latex] and add. For example, given matrices [latex]A[/latex] and [latex]B[/latex] where the dimensions of [latex]A[/latex] are [latex]2\times3[/latex] and the dimensions of [latex]B[/latex] are [latex]3\times3[/latex], the product of [latex]AB[/latex] will be a [latex]2\times3[/latex] matrix.

[latex]A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}[/latex] and [latex]B=\begin{bmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{bmatrix}[/latex]

Multiplying Matrices

Properties of Matrix Multiplication

[latex]AB=\begin{bmatrix} -7&10\\30&11\end{bmatrix} \neq \begin{bmatrix}-9&10&10\\4&-8&-12\\10&4&21\end{bmatrix}=BA[/latex]

This illustrates the fact that matrix multiplication is not commutative.

Also, this means that it is possible for [latex]AB[/latex] to be defined but not [latex]BA[/latex].  Consider a matrix [latex]A[/latex] with dimension [latex]3 \times 4[/latex] and matrix [latex]B[/latex] with dimension [latex]4 \times 2[/latex]. For the product [latex]AB[/latex], the inner dimensions are 4 and the product is defined, but for the product [latex]BA[/latex], the inner dimensions are 2 and 3, so the product is undefined.