JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Matrices

The coefficient determinant $\left|\begin{array}{ccc}2& 1& 1\\ 1& -1& 1\\ 1& 1& a\end{array}\right|=-3a+1$ vanishes when $a=\frac{1}{3}$. For $a=\frac{1}{3}$ the third row equals $\frac{2}{3}(\text{row }1)-\frac{1}{3}(\text{row }2)$, so the system is consistent only if $b=\frac{2}{3}(5)-\frac{1}{3}(3)=\frac{7}{3}$. There is no solution exactly when the determinant is zero while this consistency fails, i.e. when $a=\frac{1}{3}$ and $b\ne \frac{7}{3}$.