where the n × n matrix A has a nonzero determinant, and X, B are n × m matrices. Given sequences 1 ≤ i 1 < i 2 < … < i k ≤ n {\displaystyle 1\leq i_{1}<i_{2}<\ldots <i_{k}\leq n} and 1 ≤ j 1 < j 2 < … < j k ≤ m {\displaystyle 1\leq j_{1}<j_{2}<\ldots <j_{k}\leq m} , let X I , J {\displaystyle X_{I,J}} be the k × k submatrix of X with rows in I := ( i 1 , … , i k ) {\displaystyle I:=(i_{1},\ldots ,i_{k})} and columns in J := ( j 1 , … , j k ) {\displaystyle J:=(j_{1},\ldots ,j_{k})} . Let A B ( I , J ) {\displaystyle A_{B}(I,J)} be the n × n matrix formed by replacing the i s {\displaystyle i_{s}} column of A by the j s {\displaystyle j_{s}} column of B, for all s = 1 , … , k {\displaystyle s=1,\ldots ,k} .