Homework Help: Reducing a matrix determinant

1. May 19, 2012

Koranzite

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

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2. May 19, 2012

AKG

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

3. May 19, 2012

Koranzite

I also tried that, to no avail.

4. May 19, 2012

Ray Vickson

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