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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?

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    Thanks, that makes total sense... very neat!2011-01-25
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    I merged your two accounts. Registering will help mitigate this problem and allow you to properly comment on questions (instead of posting a comment as an answer).2011-01-25
  • 0
    You may also want to mark ulvi's answer as accepted by clicking the grey check mark to the left of the answer.2011-01-25

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