How Can You Simplify a 3x3 Matrix Determinant with Variables a and b?

[Koranzite]

Homework Statement

I've attached the problem, it involves reducing a 3x3 matrix determinant to row echelon form, but the leading diagonal elements have to be linear in a and b afterwards.

Homework Equations

The Attempt at a Solution

I've managed to convert it to row echelon form by: r3-ar2 ; r2-ar1 ; r3-br2
The problem is that this leaves a diagonal element having cubic terms. Can anyone see a way to acomplish this? Should be an easy problem, but I've spent over an hour trying different combinations.

 

[AKG]

I recommend just computing the determinant and doing some simple algebra to factor the resulting expression.

 

[Koranzite]

I recommend just computing the determinant and doing some simple algebra to factor the resulting expression.

I also tried that, to no avail.

 

[Ray Vickson]

I also tried that, to no avail.

The factorization is not immediately obvious, but what finally worked for me was to look at the determinant for two different numerical values of a (namely, a = 0 and a = 1) and in each case to factor the resulting polynomial in b. Some factors are the same for both values of a, and some others differ in such a way that you can easily figure out what they are as functions of a. You end up with a factorization exactly of the required type.

RGV