Determinant

The determinant is a scalar value that is a function of the entries of a square matrix. The formula to calculate the determinant of a square matrix $A=(a_{ij}) \in M_{n}$

$$
\det(A) = \left| A \right| = a_{11}A_{11} + a_{12}A_{12} + … + a_{1n}A_{1n}
$$
$$
A_{ik} = (-1)^{i+k}det(M_{ik})
$$

$M_{ik}$ is a square matrix $\in M_{n-1}$ that is created from A matrix by removing row $i$ & column $k$

1. $A\in M_{2}$

For example, we have a square $2\times2$ matrix:

$$
A =
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
$$
$$
\Rightarrow
\det(A) = a_{11}A_{11} + a_{12}A_{12} = a_{11}(-1)^{1+1}\left| a_{22} \right| + a_{12}(-1)^{1+2}\left| a_{21} \right| = a_{11}a_{22}-a_{21}a_{12}
$$

The matrix A is used to represent two vectors $\overrightarrow{u} = (a_{11},a_{21})$ & $\overrightarrow{v} = (a_{12},a_{22})$. The determinant of matrix A is the area of the parallelogram that is created by two vectors $\overrightarrow{u}$ & $\overrightarrow{v}$. Let’s describe these two vectors on the graph

Therefore we can calculate the area of parallelogram

\[
\begin{align}
area &= (a_{11}+a_{12})(a_{21}+a_{22})-a_{12}a_{22}-a_{11}a_{21}-2a_{12}a_{21} \\
&=a_{11}a_{21}+a_{12}a_{21}+a_{11}a_{22}+a_{12}a_{22}-a_{12}a_{22}-a_{11}a_{21}-2a_{12}a_{21} \\
&=a_{11}a_{22}-a_{12}a_{21} \\
&=\det(A)
\end{align}
\]

2. $A\in M_{3}$

For example, we have a square $3 \times 3 $ matrix:

$$
A =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
$$
$$
\begin{align}
\det(A) &=
a_{11}(-1)^{1+1}
\begin{vmatrix}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{vmatrix}
+ a_{12}(-1)^{1+2}
\begin{vmatrix}
a_{21} & a_{23} \\
a_{31} & a_{33}
\end{vmatrix}
+ a_{13}(-1)^{1+3}
\begin{vmatrix}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{vmatrix} \\
&=a_{11}(a_{22}a_{33} – a_{32}a_{23}) – a_{12}(a_{21}a_{33} – a_{31}a_{23}) + a_{13}(a_{21}a_{32} – a_{31}a_{22}) \\
&=a_{11}a_{22}a_{33} – a_{11}a_{32}a_{23} – a_{12}a_{21}a_{33} + a_{12}a_{31}a_{23} + a_{13}a_{21}a_{32} – a_{13}a_{31}a_{22} \\
&=(a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}) – (a_{11}a_{32}a_{23} + a_{12}a_{21}a_{33} + a_{13}a_{31}a_{22})
\end{align}
$$