If I tell you that each of the four numbers 1242,1426,2070,and 1564 is divisible by 23(trust me!), could you with no explicit calculation prove that the determinant of the 4x4 matrix
1 2 4 2
1 4 2 6
1 5 6 4
2 0 7 0
is also divisible by 23?

Multiply the 1st, 2^nd and 3^rd columns by 1000, 100 and 10 respectively
and add the results to the fourth column. This will not change the
value of the determinant:

1  2  4  1242
1  4  2  1426
1  5  6  1564
2  0  7  2070:

23 is then a common factor of all elements in column 4 and therefore
a factor of the determinant.

(In fact the value of the determinant is 0)

Posted by Harry on 2014-10-02 19:39:27