Nontrivial solutions to matrix equation where the determinant of the matrix is 0

Question

Let $A$ be a $n$ x $n$ matrix such that $det A = 0$. Let $x$ be a $n$ x $1$ column vector, and let 0 denote the $n$ x $1$ column vector whose elements are all 0. Does the equation $Ax = 0$ always have a nontrivial solution $x$? If no, under which conditions does a nontrivial $x$ exist, and how would I determine it?

Answer 1

Yes, an $n \times n$ singular matrix has rank $< n$, so its null space has dimension at least $1$. To find the solutions $x$, you might use Gaussian (or Gauss-Jordan) elimination.