Today, I've got task: Find all values of parameter a, when rank of matrix M equals 2, where matrix M is 3x3 and has some dependencies on a, for example: $$\begin{pmatrix} 1 & a & a^2-1\\ 1-a & 5 & 7 \\ 2a+3 & 2 & 1+a^2 \end{pmatrix}$$
It is not very complicated task, so I found determinant of matrix, which is polynom. Then I found nulls of polynom and checked if these nulls correspond to rank 2.
But what if I have super-big matrix, with noncalculating determinant, like 4x4 =) Or what if determinant equation is some kind of complicated?
So the question is: are there other ways to solve this task, which may be scalable for bigger problem?