JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Matrices

By the Cayley-Hamilton theorem, matrix A satisfies its own characteristic equation. The characteristic equation is $|A-\lambda I|=0$, which gives:



Replacing $\lambda$ with A (Cayley-Hamilton):


Comparing with $A^2+\gamma A+18I=O$:

• Coefficient of A: $-(\beta +4)=\gamma$
• Constant term: $4\beta +2\alpha =18$

Therefore, $\det (A)=4\beta +2\alpha =18$.