Often the determinant of a matrix may be written as a product of one or more factors. We can find an expression for the determinant andfactorise it, which may be long and tedious, or we may look at the matrix itself and see which changes we can make so that the matrix becomes linearly independent (if it isn't already).
Example: Find factors in the determinant of the matrix
The determinant is
(1)
If
then the matrix becomes
The first two rows are the same and the determinant is zero, so
is a factor.
If
then the matrix becomes
The first and third rows are the same and the determinant is zero, so
is a factor.
If
then the matrix becomes
The last two rows are the same and the determinant is zero, so
is a factor.
The matrix can have at most three factors (since it has three rows) so
is a factor and can differ from the determinant only by a constant factor. In fact, multiplying out the above expression gives
This differs from (1) only by a factor -1 so the determinant can be factorised as