The minor of a determinant is created by deleting all elements in the same row or column as the selected element:
#|(5,4,color(red)(cancel(color(black)2))),
(color(red)(cancel(color(black)(1))),color(red)(cancel(color(black)(7))),color(red)(cancel(3))),
(4,-1,color(red)(cancel(color(black)(1))))|
=|(5,4),(4,-1)|#
The co-factor of a determinant is created by multiplying the minor by #(-1)^(r+c)# where #r# is the row number of the selected element and #c# is the column number of the selected element.
In this case #color(red)(3)# is in row #2# column #3#.
(The top left is always row #1#, column #1#)
So for this example
#color(white)("XXX")(-1)^(r+c)=(-1)^(2+3)=(-1)^5=-1#
and the co-factor is
#color(white)(""XXX")-1 * |(5,4),(4,-1)|#
