The theory determinant is a vast topic, so we won’t dive too much into it here.

The derterminant can be used for quickly testing whether a set of vectors are linearly dependent.

## 1. Calculation

The calculation of determinant of **order 1** proceeds as follows :

The calculation of determinant of **order 2** proceeds as follows :

The calculation of determinant of **order 3** proceeds as follows :

The calculation of determinant of order higher than 3 is beyond the scope of the this course. Please refer here if you want to know more about it.

## 2. Dependence test

The test of linear dependence is done by using the following property of determinants :

The contraposition of $(4)$ gives :

#### Example I

Let’s test the linear dependence of the following vectors : $ \left(\begin{smallmatrix} 8 \\ -4 \\ -3 \end{smallmatrix}\right), \left( \begin{smallmatrix} 2 \\ -4 \\ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} -2 \\ 0 \\ 1 \end{smallmatrix} \right)$

According to $(4)$ the vetors are linearly dependent.

## 3. But more concretely ?

How does the determinant help telling about the linear dependance of vectors ?

The determinant actually gives the **area** of the parallelogram that two vectors of $\mathbb{R}^2$ (Figure 7.1) form together, respectively the **volume** of the parallelepiped that three vectors of $\mathbb{R}^3$ form togheter (Figure 7.2).

Figure 7.3 illustrates the line segment that two linearly dependent vectors of $\mathbb{R}^2$ form together. Since that one has no width, the area is equal to $0$. Figure 7.4 illustrates the same idea in $\mathbb{R}^3$. Without any height the volume equals $0$.

## Recapitulation

**Determinant** of vectors can be used for quickly testing whether those vectors are linearly dependent or not.

The determinant gives (in absolute value) the **area** of the parallelogram that two vectors of $\mathbb{R}^2$ form together, respectively the **volume** of the parallelepided that three vectors of $\mathbb{R}^3$ form together. More generally, the determinant gives the hypervolume of the hyperparallelepiped that $n$ vectors of $\mathbb{R}^n$ form together.

If a set of vectors are linearly dependent, then their determinant equals $0$.