A) \[-18\]
B) \[54\]
C) \[-72\]
D) \[72\]
Correct Answer: C
Solution :
Given, \[M=\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right)\] Then, \[{{M}^{T}}=\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right)\] \[3(M+{{M}^{T}})=3\left[ \left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right)+\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right) \right]=3\left( \begin{matrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{matrix} \right)\] \[=\left( \begin{matrix} 0 & 0 & 6 \\ 0 & 6 & 0 \\ 6 & 0 & 0 \\ \end{matrix} \right)\] det \[(3(M+{{M}^{T}}))\left| \begin{matrix} 0 & 0 & 6 \\ 0 & 6 & 0 \\ 6 & 0 & 0 \\ \end{matrix} \right|=6\left| \begin{matrix} 0 & 6 \\ 6 & 0 \\ \end{matrix} \right|\] \[=6(0-36)=6\times (-36)\] Now, \[\frac{1}{3}\det \,(3(M+{{M}^{T}}))=\frac{1}{3}\times 6\times (-36)\] \[=2\times (-36)=-72\]
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