16.1 Formula for $|A|$
For a $2 \times 2$ matrix $A$, the formula for $|A|$ can be derived as follows:
$$\begin{align} |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix}= \begin{vmatrix} a & 0 \\ c & d \end{vmatrix}+\begin{vmatrix} 0 & b \\ c & d \end{vmatrix} \end{align}$$
$$\begin{align} =\begin{vmatrix} a & 0 \\ c & 0 \end{vmatrix}+\begin{vmatrix} a & 0 \\ 0 & d \end{vmatrix}+\begin{vmatrix} 0 & b \\ c & 0 \end{vmatrix}+\begin{vmatrix} 0 & b \\ 0 & d \end{vmatrix} \end{align}$$
$$\begin{align} =0+ad-bc+0=ad-bc \end{align}$$
For a $3 \times 3$ matrix, first row can be seperated into $3$ pieces as demonstrated above. For each of the individual separated matrices, the second row will be separated into $3$ pieces giving a total of $9$ matrices. Finally, for each of these $9$ matrices, the third row will be separated into $3$ pieces, giving a total of $27$ matrices. Out of these $27$ matrices, a lot will have $0$ determinant. Matrices with non-zero determinant will have one entry from each row and column. The splitted matrices with non-zero determinant for a $3 \times 3$ matrix $A$ is shown below. The sign of individual determinants is derived based on number of row exchanges needed to get a diagonal matrix.
$$\begin{align} |A| = \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} \end{align}$$
$$\begin{align} =\begin{vmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33} \end{vmatrix}+\begin{vmatrix} a_{11} & 0 & 0\\ 0 & 0 & a_{23}\\ 0 & a_{32} & 0 \end{vmatrix}+\begin{vmatrix} 0 & a_{12} & 0\\ a_{21} & 0 & 0\\ 0 & 0 & a_{33} \end{vmatrix} \end{align}$$
$$\begin{align} +\begin{vmatrix} 0 & a_{12} & 0\\ 0 & 0 & a_{23}\\ a_{31} & 0 & 0 \end{vmatrix}+\begin{vmatrix} 0 & 0 & a_{13}\\ a_{21} & 0 & 0\\ 0 & a_{32} & 0 \end{vmatrix}+\begin{vmatrix} 0 & 0 & a_{13}\\ 0 & a_{22} & 0\\ a_{31} & 0 & 0 \end{vmatrix} \end{align}$$
$$\begin{align} =a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}+ \\ a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} \end{align}$$
The generic formula for a $n \times n$ matrix $A$ is as follows:
$$\begin{align} |A| = \sum_{\text{n! terms}} \pm a_{1\alpha}a_{2\beta}…a_{n\omega} \end{align}$$
$$\begin{align} \text{such that}: (\alpha, \beta, …, \omega) = \text{Permutation of}(1,2,3,…,n) \end{align}$$
16.2 Cofactors
Cofactor Formula connects determinant of $n \times n$ matrix to the determinant of smaller matrix of size $n-1 \times n-1$. The determinant of the $3 \times 3$ matrix $A$ shown in cofactor format is as follows:
$$\begin{align} |A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) + a_{12}(-a_{21}a_{33} + a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \end{align}$$
The above determinant formula can be seen as the combination of following permutations.
$$\begin{align} \begin{vmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & a_{23}\\ 0 & a_{32} & a_{33} \end{vmatrix};\begin{vmatrix} 0 & a_{12} & 0\\ a_{21} & 0 & a_{23}\\ a_{31} & 0 & a_{33} \end{vmatrix};\begin{vmatrix} 0 & 0 & a_{13}\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & 0 \end{vmatrix} \end{align}$$
$$\begin{align} \text{Cofactor of } a_{ij} = C_{ij} = (-1)^{i+j}|n-1 \text{ matrix with row } i \text{ col } j \text{ erased}| \end{align}$$
Cofactors without the sign are called Minors. Hence,
$$\begin{align} |A| = a_{11}C_{11} + a_{12}C_{12} + … + a_{1n}C_{1n} \end{align}$$