JEE Main 2026 — 22 January, Morning Shift — Question 13

$\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\3 & 5\end{array}\right]$

$\Rightarrow \mathrm{A}^{2}=\left[\begin{array}{ll}13 & 21 \\21 & 34\end{array}\right]$

$\left|\mathrm{A}^{2025}-3 \mathrm{~A}^{2024}+\mathrm{A}^{2023}\right|$

$=\left|\mathrm{A}^{2023}\left(\mathrm{~A}^{2}-3 \mathrm{~A}+\mathrm{I}\right)\right|$

$=|\mathrm{A}|^{2023}\left|\mathrm{~A}^{2}-3 \mathrm{~A}+\mathrm{I}\right|$

$=1 .\left|\begin{array}{cc}8 & 12 \\12 & 20\end{array}\right|=160-144=16$