We will define and understand Leibnitz Method in order to find the determinant of a square matrix and also follow this by a worked example.

Definition (Leibnitz Expansion): Let {\bf A}=(a_{ij}) be an n\times n square matrix. Then \det (A) = \sum_{\sigma\in S_n}\text{sgn} (\sigma) \Pi_{i=1}^{n} a_{i \sigma_i}.

Leibnitz method of finding the determinant of an n\times n matrix depends on all the permutations of the set \lbrace 1,2,\cdots,n\rbrace. In fact, the set S_n denotes the set of all permutations of 1,2,\cdots,n . A permutation \sigma\in S_n is a bijective function from \lbrace 1,2,\cdots,n\rbrace to itself.

Exercise 1: Find all the permutation of \lbrace 1,2,3\rbrace. Hence construct the set S_3.

Example: Let us find the determinant of the matrix A=\begin{pmatrix}1 & 3 & 4\\ 5 & 2 & 3\\ 6 & 1 & 3\end{pmatrix} using Leibnitz expansion.

First of all for a 3\times 3 matrix, the Leibnitz expansion becomes: \det (A)=\sum_{\sigma\in S_3}\text{sgn} (\sigma) a_{1 \sigma_1}a_{2 \sigma_2}a_{3 \sigma_3}