Given is an almost-square matrix $A$ with $n$ columns and $n-1$ rows with maximum rank. The solutions of the homogeneous system $Ax = 0$ form a 1-dimensional subspace of $\mathbb{R}^n$.
I've discovered the following which I believe to be true but I can't prove: the components of the vector $x$ that spans the (1D) solution space are given by:
$x_i = (-1)^{i-1} |A_i|$
in which $|A_i|$ is the determinant of the square submatrix of A obtained by removing the i-th column from A. For example, in $\mathbb{R^3}$, $A$ is a 2x3 matrix, and $x$ as defined above turns out to be the crossproduct of the two row vectors of $A$.
Is this true, and if so, how can it be proved?