A) \[\text{det}\,(adj\,\,{{\text{M}}^{2}})\,=\,81\]
B) \[a+b=3\]
C) \[(adj\,{{\text{M}}^{-1}})\,+\,adj\,\,{{\text{M}}^{-1}}=\,-\text{M}\]
D) \[\text{M}\left[ \begin{matrix} \alpha \\ \beta \\ \gamma \\ \end{matrix} \right]\,=\,\left[ \begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix} \right]\,,\,\text{the}\,\,\alpha \,-\,\beta +\,\gamma \,=\,3\]
Correct Answer: B , C , D
Solution :
\[\text{M}=\left[ \begin{matrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \\ \end{matrix} \right]\] and adj \[\text{M}=\left[ \begin{matrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \\ \end{matrix} \right]\]\[\Rightarrow \]adj\[\text{M}=\left[ \begin{matrix} 2-3b & ab-1 & -1 \\ 8 & -6 & 2 \\ b-6 & 3 & -1 \\ \end{matrix} \right]\]\[=\left[ \begin{matrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \\ \end{matrix} \right]\] |
\[2-3b=-1;ab-1=1\] |
\[b-6=-\,5;a=2\] |
\[b=1\] |
Now, \[\text{M}=\left[ \begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \\ \end{matrix} \right]\] |
\[\left| \text{M} \right|=8-10=-2\] |
\[\Rightarrow \] \[a+b=3\]option [b] |
\[\left| \text{adj}\,({{\text{M}}^{2}}) \right|={{\left| {{\text{M}}^{2}} \right|}^{2}}\] \[={{\left| \text{M} \right|}^{4}}=16\] |
[c] \[{{\text{(adj}\,\text{M})}^{-1}}+\text{adj}\,({{\text{M}}^{-1}})\,\text{option}\,(c)\]\[=\,\text{adj}\,({{\text{M}}^{-1}})+\text{adj}\,({{\text{M}}^{-1}})\]\[=2\text{adj}\,({{\text{M}}^{-\,\,1}})\]\[=2(\left| {{\text{M}}^{-\,\,1}} \right|\text{M})=2\left( \frac{1}{-2}\text{M} \right)=\,-\,\text{M}\] |
[d] \[\text{M}\left[ \begin{matrix} \alpha \\ \beta \\ \gamma \\ \end{matrix} \right]\,=\,\left[ \begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix} \right]\] |
\[\left[ \begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} \alpha \\ \beta \\ \gamma \\ \end{matrix} \right]\,=\,\begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix}\] |
\[\beta +2\gamma =1\] |
\[\alpha +2\beta +3\gamma =2\] |
\[3\alpha +\beta +\gamma =1\] |
\[\alpha =1,\]\[\beta \]= \[-1,\]\[\gamma =1\] |
\[\alpha -\beta +\gamma =3\]Option [d] |
Solution :
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