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Can I assume that an equation (with 3 variables) is unsolvable if its determinant equals zero.Sure, it not true if all it's other determinants are zero as well, but it seems to be unlikely

it is true for a determinant such as (1,0,0),(0,1,0),(0,0,1) but thats not a real equation.

if this is the case is there an easy way to prof that? otherwise could anyone provide an example with for which this rule doesn't work.

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    as Manos said, I'm referring to Cramer's Rule2012-10-29

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What you are referring to is known as "Cramer's Rule". When we have an $n \times n$ linear system of equations $Ax=b$ and the determinant of the coefficient matrix $A$ is nonzero (equivalently $A$) is invertible, then we can apply Cramer's rule, which involves divisions with $det(A)$ to obtain the unique solution $A^{-1}b$. It might be possible though that $det(A)=0$, in which case Cramer's Rule does not apply, and still the system has a solution. The correct criterion to check is whether $b \in \mathcal{R}(A)$, i.e. the right hand side vector must lie in the range space of $A$, i.e. we can construct $b$ as a linear combination of columns of $A$. A simple example is the system $\left[\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right] x = \left[\begin{array}{c} 3\\ 0 \end{array} \right]$. Then every vector of the form $\left[\begin{array}{c} 3\\ \alpha \end{array} \right], \, \alpha \in \mathbb{R}$, is a solution.

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    In my example i am using the real number field $\mathbb{R}$. I am not aware of the significance of the other determinants appearing in Cramer's rule in the nonexistence of a solution, but since determinants are hard to compute, it is better to use the criterion $rank[A |b]=rank[A]$. A solution exists if and only if the above equality is true. Checking this is very easy, e.g. if you are using MATLAB via the "rank" function. Do you know what rank is?2012-10-29