Determinant of Matrices

Definition

Let $A$ : Square matrix of order $n$

Let

• $a_{ij}$ : Element of $A$ at row $i$ column $j$
• $M_{ij}$ : $(n-1)\times (n-1)$ square submatrix obtained by deleting row $i$ and column $j$ of $A$

Choose any $i\in \{1,\dots ,n\}$

Then

$$\det (A)=\sum _{j=1}^n(-1{)}^{i+j}{a}_{ij}\det (M_{ij})$$

Note

You can choose $j\in \{1,\dots ,n\}$, and sum over $i=1,\dots ,n$ instead