JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Matrices

With $|A|=2$ and $\mathrm{trace}(A)=-3$, the Cayley–Hamilton theorem gives

so $x=3,y=2$ is the only point satisfying $A^2+xA+yI=0$. A single point does not determine a hyperbola, so its eccentricity and latus rectum are not fixed — Allen marks the question BONUS; the NTA key gives $25$. The stored value $9999$ is a sentinel placeholder.