# 5.3. Matrices: Matrices & simultaneous linear equations

### Learning Objectives

You should be able to

• translate between a system of $$n$$ simultaneous linear equations in $$m$$ unknowns and a matrix equation involving an $$n\times m$$ coefficient matrix and also write down the associated $$n \times (m+1)$$ augmented matrix
• Solve small systems of linear equations using matrix inverses or Gaussian elimination on the augmented matrix

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.

Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 5.3.1 Encoding systems into matrix equations

$$\begin{cases} ax+by=c\\ dx+ex=f\end{cases} \iff \begin{pmatrix} a & b\\ d & e \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}=\begin{pmatrix} c\\ f \end{pmatrix}$$
coefficient matrix is $$\begin{pmatrix} a & b\\ d & e \end{pmatrix}$$,      augmented matrix is $$\left( \begin{matrix} a & b\\ d & e \end{matrix} \middle| \begin{matrix} c\\ f \end{matrix} \right)$$

#### 5.3.2 Solving simultaneous linear equations using matrices

See examples of how to solve simultaneous linear equations using matrices by either

• multiplying by a matrix inverse or
• performing Gaussian elimination on the augmented matrix.

### Economic Application: Labour market transitions

This is the last, of a sequence of applications, explaining how matrices can be used to represent the dynamic change in a labour market, in terms of the distribution of workers across labour market states (employment, unemployment, non-participation). In particular, it is demonstrated how given the state of the economy in a given period of time, one can obtain the state in the previous period by using the inverse of the transition matrix.

#### Economic Application Exercise

The following quiz allows you to test your understanding of using the inverse of the transition matrix to obtain the state of the economy in a previous period, in a simplified model of a labour market with two states only.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to. If you want to test yourself further, then try the advanced quiz linked below.