Computing first-order and second-order determinants

A matrix $1\times 1$ is just a number. Its determinant is that same number.

Let's see how to calculate the determinant of a $2\times 2$ matrix

$$\left| \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right|=2 \cdot 3-1\cdot 5=1\\\left| \begin{matrix} 0 & 4 \\ 2 & 1 \end{matrix} \right|=0 \cdot 1-4\cdot 2=-8$$

Can you calculate now the following determinant?

$$\left| \begin{matrix} 1& 3 \\ 0 & 5 \end{matrix} \right|=?$$

Indeed, looking at the examples above, you can see

$$\left| \begin{matrix} 1& 3 \\ 0 & 5 \end{matrix} \right|=1 \cdot 5-3 \cdot 0=5$$

What about this?

$$\left| \begin{matrix} 1& 4 \\ 2 & 8 \end{matrix} \right|=?$$

$$\left| \begin{matrix} 1& 4 \\ 2 & 8 \end{matrix} \right|=1 \cdot 8- 2 \cdot 4=0$$

We are going to do it generally, whatever the matrix elements are. In this case,

$$\left|\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} \right|=a_{11}\cdot a_{22}- a_{21}\cdot a_{12}$$