Matrix Determinant

Compute Matrix Determinant
A square matrix $A$, $$\begin{array}{r}A=\left[\begin{array}{cccc}{a}_{1,1}& a_{1,2}& ...& a_{1,n}\\ a_{2,1}& a_{2,2}& ...& a_{2,n}\\ ⋮& ⋮& ⋮& ⋮\\ a_{n,1}& a_{n,2}& \dots & a_{n,n}\end{array}\right]\end{array}$$ Be $A$ a matrix $n\times n$ and $n⩾2$. The determinant of $A$ is scalar give by $|A|=det(A)$, $$det(A)=\sum _{k=1}^n{a}_{i,j}{C}_{i,j}$$ where $C_{i,j}$ is cofactor, $$C_{i,j}=(-1{)}^{i+j}det(A_{i,j})$$ Example $\bullet$ Compute the determinant of matrix $A$ showed bellow, $$\begin{array}{r}A=\left[\begin{array}{ccc}1& 1& 3\\ 2& 5& 7\\ 12& 6& 1\end{array}\right]\end{array}$$ Resolution Choice expansion for row $1$, $$det(A)=\sum _{k=1}^3{a}_{1,k}(-1{)}^{1+k}det\left(A_{1,k}\right)=$$
$$=1(-1{)}^{1+1}det\left(\left[\begin{array}{cc}5& 7\\ 6& 1\end{array}\right]\right)+1(-1{)}^{1+2}det\left(\left[\begin{array}{cc}2& 7\\ 12& 1\end{array}\right]\right)+3(-1{)}^{1+3}det\left(\left[\begin{array}{cc}2& 5\\ 12& 6\end{array}\right]\right)=$$
$$=1\cdot (5\cdot 1-7\cdot 6)+(-1)(2\cdot 1-7\cdot 12)+3\cdot 1\cdot (2\cdot 6-5\cdot 12)=$$
$$=-37+82-144=\boxed{{\displaystyle -99}}$$

PDF Text: Determinant of Matrix A