Let K {\displaystyle {}K} be a field, and let M {\displaystyle {}M} denote an n × n {\displaystyle {}n\times n} -matrix over K {\displaystyle {}K} with entries a i j {\displaystyle {}a_{ij}} . For i ∈ { 1 , … , n } {\displaystyle {}i\in \{1,\ldots ,n\}} , let M i {\displaystyle {}M_{i}} denote the ( n − 1 ) × ( n − 1 ) {\displaystyle {}(n-1)\times (n-1)} -matrix, which arises from M {\displaystyle {}M} , when we remove the first column and the i {\displaystyle {}i} -th row. Then one defines recursively the determinant of M {\displaystyle {}M} by
be a
field,
and let 
denote an
-matrix
over with entries 
. For
,
let 
denote the 
-matrix, which arises from , when we remove the first column and the 
-th row. Then one defines recursively the determinant of by
