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

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

Can you calculate now the following determinant?

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

Indeed, looking at the examples above, you can see

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

What about this?

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

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

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

$$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=a_{11}\cdot a_{22}-a_{21}\cdot a_{12}$$