# 5.1. Matrices: Matrix arithmetic

### Learning Objectives

You should be able to

• understand and use notation and terminology relating to matrices
• understand when you can add, subtract or multiply two matrices together and be able to perform the calculation
• multiply a matrix by a scalar or a vector

### 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

#### 5.1.1. Matrix terminology & notation

$$n\times m$$ matrix has $$n$$ rows and $$m$$ columns,     $$A_{i,j}$$ is entry in row $$i$$ and column $$j$$,
$$A'=A^T$$ is the transpose.      $$|A| = \det A$$ is the determinant,       $$I_n$$ is the identity matrix,   $$0_{n\times m}$$ is the zero matrix,
$$\mathrm{tr}(A)$$ is the trace,    $$A^{-1}$$ is the inverse

#### 5.1.2. Matrix addition & scalar multiplication

Matrix addition and scalar multiplication are component-wise.

#### 5.1.3. Matrix multiplication

$$AB$$ is defined if $$A$$ is $$n\times m$$ and $$B$$ is $$m \times k$$. The result $$AB$$ is $$n\times k$$.
To find the $$(i,j)^{th}$$ entry of $$AB$$, take the scalar product of the $$i^{th}$$ row of $$A$$ and the $$j^{th}$$ column of $$B$$.

### Economic Application: Labour market transitions

This is the first, 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 the complex transitional dynamics in the market can be represented succinctly by the means of a dynamic equation involving matrices. The discussion requires conceptual understanding of matrices and matrix multiplication.

#### 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.

### 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.