The question, pure curiosity, is whether you can solve a quadratic with the use of matrices?
And if yes, does that method also work for higher polynomials?
Say for example I have a quadratic such as written below: \begin{equation} x^2+3x+2=0 \end{equation} By simple other known methods, one ends up with $x = -1$ or $-2$.
(I already know it is possible to find the solution of a system of simultaneous equations... Maybe does that help?)
Can you get to those results with matrices? Thanks a lot for your contribution and answers!
Here's one way to do use matrices here: we are given the problem $$ u^2 + 3u + 2 = 0 $$ substituting $u = x/y$, we have $$ x^2/y^2 + 3x/y + 2 = 0 \implies\\ x^2 + 3xy + 2y^2 = 0 \implies\\ \pmatrix{x&y} \pmatrix{1&3/2\\3/2&2} \pmatrix{x\\y} = 0 $$ this is now a problem about quadratic forms, which can be solved using knowledge about symmetric/Hermitian matrices.
This particular method doesn't extend to higher degree polynomials.
Another approach: the zeros of a polynomial are precisely the eigenvalues of the associated companion matrix. This works for polynomials of any degree.
In your particular case, we can find the zeros of $x^2 + 3x + 2$ by finding the eigenvalues of the matrix $$ \pmatrix{0&-2\\1&-3\\} $$