6.02 Matrix equations and applications

Equations involving matrices

Two matrices are said to be equal if every corresponding elements in the matrices are equal.

So in this case, since the two matrices are equal, then $m=3$m=3 and $n=7$n=7.

Simultaneous equations

Simultaneous equations follow the convention of organising the coefficients into matrix notation. For instance, consider the two equations $x+2y=7$x+2y=7and $2x-5y=-4$2x−5y=−4. This system can be written like this using matrices.

This will be covered in more detail when looking at matrix multiplication in a later lesson.

Practice questions

Question 1

Using matrices to model practical examples

Writing information in matrix form

When simply being used to store data, is it not necessary to use the labelling conventions and simply create the matrix. Consider the following table:

The above situation is converted to a matrix like this: 

However, when undertaking mathematical processes using the data stored in matrices, the defining of rows and columns must follow normal conventions.  

Two-way frequency tables

Two-way frequency tables follow the convention of independent variable headings in the columns and dependent variables as the rows.  

A survey was completed at a school that has both secondary and primary school students.  They were asked if the playground should have fewer, same or more seating options. The results are displayed in the table below: 

 This can be written as a matrix.  The result would be: 

Worked examples

Example 1

Here is a network showing roads between towns. Put the information into a matrix.

Think: We can construct a matrix, where the entries represent the number of paths between the towns.

Do: We will need to set up a $5\times5$5×5 matrix, where each of the rows and columns represents a town.

The next step is to fill in the numbers of roads between them. There are two roads between towns $1$1 and $3$3, so we input a $2$2 in $a_{13}$a13​ and $a_{31}$a31​. As there are no roads connecting town $2$2 with any other towns, all elements in row $2$2 and column $2$2 are $0$0.

Example 2

Generate a matrix to represent the following two way frequency table for party preference for the over $30$30's and under $30$30's. Let the rows represent the values of the party preferences and the columns represent the age groups.

Think: Since the rows and columns in the two way frequency table correspond to the rows and columns of the matrix, we can simply transfer the information into a matrix.

Do: The matrix is then given by the following:

Practice questions

Question 2