JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Matrices

The coefficient determinant is $\Delta =\left|\begin{array}{ccc}2& -1& 3\\ 3& 2& -1\\ 4& 5& \alpha \end{array}\right|=7(\alpha +5)$. When $\alpha \ne -5$, $\Delta \ne 0$, so the system has a unique solution for every value of $\beta$. When $\alpha =-5$, $\Delta =0$ and the behaviour depends on $\beta$: replacing the constants column gives ${\Delta }_3=7(\beta -9)$, so $\beta =9$ produces infinitely many solutions while $\beta =8$ makes the system inconsistent. For $\alpha =-6$, $\Delta =7(-1)=-7\ne 0$, so the system then has a unique solution rather than infinitely many; the statement asserting infinitely many solutions when $\alpha =-6,\beta =9$ is therefore the one that is not correct.