Laplace expansion

Laplace expansion, also known as the cofactor expansion, is a method used in linear algebra to compute the determinant of a square matrix. This technique breaks down the determinant into a sum of determinants of smaller matrices, making it a recursive process.
Specifically, for every \[i\], the Laplace expansion along the \[i\]th row is equal to:
\[\det(B)=\sum^{n}_{j=1}(-1)^{i+j}b_{i,j}m_{i,j}\]
Similarly, the Laplace expansion along the \[j\]th column is equal to:
\[\det(B)=\sum^{n}_{i=1}(-1)^{i+j}b_{i,j}m_{i,j}\]

Which means, any row or column we choose to perform this calculation will yield us the same \[\det(B)\], as the determinant of a matrix and its transpose is the same. Here, \[b_{i,j}\] refers to the element at position \[(i,j)\] in the matrix \[B\] and \[m_{i,j}\] refers to the minor of \[b_{i,j}\] (i.e. the determinant of the submatrice formed by removing the \[i\]-th row and \[j\]-th column from \[B\]).

Consider the matrix \[B\]:

\begin{align*} B= \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix} \end{align*} \begin{align*} \det(B)&=1\cdot \det\begin{pmatrix} 5 & 6 \\ 8 & 9 \\ \end{pmatrix} -2\cdot \det\begin{pmatrix} 4 & 6 \\ 7 & 9 \\ \end{pmatrix} +3\cdot \det\begin{pmatrix} 4 & 5 \\ 7 & 8 \\ \end{pmatrix}\\ &=1\cdot(-3)-2\cdot(-6)+3(-3)\\ &=0 \end{align*}