Determinant Factorisation

[Lunat1c]

Is there a good method or any particular way you guys use to factorise determinant to avoid staring at a problem for minutes without coming up with any answers?

Lets say I want to factorize $$\begin{bmatrix} 1 & a & a+1 \\ a+1 & 1 & a \\ a & a+1 & 1 \end{bmatrix}$$

Where should I start? Is there a general way to do this so that the process is a bit quick especially for when it comes to exams? Thanks!

 

[Office_Shredder]

Factorize a determinant? You mean expansion by rows? It's only a 3x3 matrix, just go along the first row and do it. Lots of practice will make it go by very quickly, once you get used to the pattern of how to calculate it

 

[Count Iblis]

You can try to find values for which the deteminant is zero. Another fast method is to insert some random integer for a and then compute the determinant. Note that computing a determinant of a matrix containing only integers is much less work than the determinant of a matrix contaning undetermined variables.

You know that the determinant is a third degree polynomial in a. If that polynomial has a simple factorization, then that means that the number you found must also factorize in at least 3 prime factors. Often that's enough to guess the correct formula. You then verify that by trying 3 more values of a. If the verification succeeds, you have rigorously proved the formula you guessed.


If guessing does not succeed, you can just evaluate the determinant for 4 different values of a and then do Lagrange interpolation to write down the 3rd degree polynomial.