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Factorising Determinants

Many systems of simultaneous equations have determinants and these are reflected in the matrices of coefficients. Sometimes the determinants of the matrices can be factorised by considering the symmetries of the rows (or columns).

Example

Obviously there is symmetry among the rows and if a, b and c are equal the determinant is zero. This might lead us to think that the determinant includes factors of the formIfthenThis gives us the matrix

with determinant

Then a=0, -c or c.

We may be better to consider consider the third row as the sum of the first two. This givesand

Since the third row is the sum of the first two the determinant is zero.

The determinant must be

The determinant of the original matrix is +1 and so is the coefficient of (1) soand thedeterminant is