NIMCET 2013 — Mathematics PYQ

The characteristic equation is given by $|A-\lambda I|=0$.

Expanding along the first row:

Multiplying by $-1$:

Step 2: Apply Cayley-Hamilton Theorem

Replace $\lambda$ with the matrix $A$:

Step 3: Rearrange to match the given form

The given equation involves $A^{-1}$, so we multiply the entire equation by $A^{-1}$:

Rearrange to isolate $6A^{-1}$:

Step 4: Compare with the given equation

The problem gives us:

By comparing the coefficients:

• $c=-6$

• $d=11$

Therefore, $(c,d)=(-6,11)$.

Correct Option: 1. (-6, 11)