# 5.2. Matrices: Determinants & inverses

### Learning Objectives

You should be able to

• Compute the determinant of a $$2\times 2$$ matrix
• Determine when a $$2\times 2$$ matrix is invertible by using the determinant
• Compute the inverse of a $$2\times 2$$ matrix when it exists

### 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.
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

#### 5.2.1 Determinant of a $$2\times 2$$ matrix

$$\begin{vmatrix}a & b \\c& d \end{vmatrix} =\det \begin{pmatrix}a & b \\c& d \end{pmatrix}= ad-bc$$

#### 5.2.2 Inverse of a $$2\times 2$$ matrix

$$\begin{pmatrix}a & b \\c& d \end{pmatrix} = \dfrac{1}{ad-bc} \begin{pmatrix}d & -b \\-c& a \end{pmatrix}$$

### Economic Application: Labour market transitions

This is the second, 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 one can obtain the determinant and the inverse of the transition matrix. The discussion requires conceptual understanding of determinants and inverses.

#### Economic Application Exercise

The following quiz allows you to test your understanding of using matrices to represent dynamic change in a simplified model of a labour market with two states only, and to compute the determinant and inverse of the transition matrix.

### 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.