PYQ (JEE Main & Advanced)

Essential Concepts for JEE Main & Advanced

For JEE Main and JEE Advanced problems on determinants and matrices, quick identification of the correct property is often more important than long expansion.

1. High-Yield Adjoint Properties

Let $A$ be a square matrix of order $n$.

• $|\operatorname{adj} A| = |A|^{n-1}$
• For a non-singular matrix, $\operatorname{adj}(\operatorname{adj} A) = |A|^{n-2}A$
• Therefore, $$|\operatorname{adj}(\operatorname{adj} A)| = |A|^{(n-1)^2}$$
• $\operatorname{adj}(kA) = k^{n-1}\operatorname{adj} A$
• $\operatorname{adj}(AB) = (\operatorname{adj} B)(\operatorname{adj} A)$
• $|kA| = k^n|A|$

2. Systems of Linear Equations

For a system $AX=B$:

• Unique solution: $|A| \neq 0$
• No solution: $|A| = 0$ and at least one of Cramer's determinants is non-zero, or equivalently $(\operatorname{adj} A)B \neq O$
• Infinitely many solutions: $|A| = 0$ and all the corresponding determinants vanish, or equivalently $(\operatorname{adj} A)B = O$

3. Homogeneous System

For $AX=O$:

• Only trivial solution: $|A| \neq 0$
• Non-trivial solutions exist: $|A| = 0$

4. Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic equation. This is especially useful in JEE problems for:

• finding higher powers of a matrix,
• reducing polynomials in matrices,
• expressing $A^{-1}$ as a polynomial in $A$.