Calculate the co-factor of the element 2:

 |1  2  3|                                                              
 |3  4  5|                                                             
 |5  6  7|

1. Create the 2×2 MINOR determinant of the element 2.

Make the 2×2 determinant by choosing the 4 elements which are NOT in the
same row and NOT in the same column with 2.  That 2×2 MINOR determinant is

Minor of 2 = |3  5|
             |5  7|

2. Give that minor determinant its correct sign:

Use this "checkerboard" sign-scheme 3x3 matrix to decide whether to assign 
the minor determinant a + sign or a - sign:

 |+  -  +|                                                              
 |-  +  -|                                                             
 |+  -  +|

We see that in the original determinant

 |1  2  3|                                                              
 |3  4  5|                                                             
 |5  6  7|

the 2 in in the middle of the top row and we see that in the sign-scheme
determinant above, there is a - sign in the middle of the top row.  Therefore
we give the minor determinant a negative sign, and the cofactor is


                     |3  5|
(-1) times the minor |5  7| = (-1)·(3·7 - 5·5) = -(21 - 25) = -(-4) = 4 

So the cofactor of 2 in the original determinant is 4.

Edwin