My teacher told me, that there exists a system of $3$ equations with $3$ unknowns, which isn't indeterminate despite all determinants being equal to $0$. Is it true? How to find this system?
2026-03-25 07:43:56.1774424636
System of equations for which Cramer's rule is invalid
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Yes, this can happen when the system has no solution and there are at least three indeterminates. For instance, for $$\begin{cases} x+y+z=1\\ x+y+z=2\\ x+y+z=3\end{cases}$$ all four determinants that appear in Cramer's rule are zero. This can't quite happen in a $2\times 2$ system (unless the matrix of coefficients is zero) because of Rouché-Capelli and the scarcity of columns.