question_answer

 \[{{A}^{-1}}=\frac{1}{6}[{{A}^{2}}+cA+dI]\] where \[c,\text{ }d\in \text{ }R,\]then pair \[(c,d)=\]

A) \[(6,11)\]              

B)   \[(6,-11)\]

C) \[(-6,11)\]              

D)   \[(-6,-11)\]

Correct Answer: C

Solution :

Given that matrix A of order \[3\times 3\] and I is the identify matrix of order \[3\times 3\]

and

Also given, \[{{A}^{-1}}=\frac{1}{6}({{A}^{2}}+cA+dI)\]

\[\Rightarrow \,\,A{{A}^{-1}}=\frac{1}{6}({{A}^{3}}+c{{A}^{2}}+dA)\]

\[\Rightarrow \,\,I=\frac{1}{6}({{A}^{3}}+c{{A}^{2}}+dA)\]

\[\Rightarrow \,\,6I={{A}^{3}}+c{{A}^{2}}+dA\Rightarrow {{A}^{3}}+c{{A}^{2}}+dA-6I=O\]  ?(i)

Now,

and

Putting these values in eq. (i),

So, we have five equations and four unknowns.

On comparing,

\[1+c+d-6=0;\]  \[-11-c+d-6=0;\]

\[19+5c+d=0;\] \[-38-10c-2d=0\]

\[46+14c+4d-6=0\]

On, solving, we get \[c=-6,\] \[d=11.\]