If a system of linear equations have non trivial solutions according to Cramer's Rule (i.e. infinitely many solutions) then it means that zero is also one of it's solutions (since it has INFINITELY many solutions). Now zero is a trivial solution. So having non trivial solutions means that it also has a trivial solutions?
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1$\begingroup$ Are you talking about homogeneous linear systems ($Ax=0$) or linear systems in general ($Ax=b$)? $\endgroup$– nathan.j.mcdougallJul 1, 2020 at 5:18
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The system $Ax=0$ always has the trivial solution, and $Ax=b$ when $b≠0$ does not. Having an infinite number of solutions does not necessarily mean that $0$ is one of them; consider the system:
$A=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $b=[1,0]$
Every $x=[y,1]$ (for every $y$) solves $Ax=b$, thus you have infinite solutions. However $x=[0,0]$ is not a solution.
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$\begingroup$ Can non homogeneous system of equation ever have zero solution? $\endgroup$ Jul 1, 2020 at 14:41
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$\begingroup$ Oh ok. When you said, "Having an infinite number of solutions does not NECESSARILY mean that 0 is one of them", I thought you meant that 0 is not necessarily included in infinitely many solutions, but it could possibly be. Anyways, really thanks for the explanation. $\endgroup$ Jul 1, 2020 at 16:13