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)|#