A) \[diag(-4,\,\,3,\,\,18)\]
B) \[diag(5,\,\,4,\,\,11)\]
C) \[diag(3,\,\,1,\,\,8)\]
D) None of these
Correct Answer: A
Solution :
Key Idea If \[A=diag\,\,({{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}})\] and\[B=diag({{b}_{1}},\,\,{{b}_{2}},\,\,{{b}_{3}})\] then\[AB=diag({{a}_{1}},\,\,{{b}_{1}},\,\,{{a}_{2}},\,\,{{b}_{2}},\,\,{{a}_{3}},\,\,{{b}_{3}})\] Given matrix can be rewritten as \[A=\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \\ \end{matrix} \right]\]and\[B=\left[ \begin{matrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \\ \end{matrix} \right]\] Now,\[{{A}^{2}}=\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \\ \end{matrix} \right]\] \[\Rightarrow \] \[\left[ \begin{matrix} -4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 18 \\ \end{matrix} \right]\] \[=diag\,\,(-4,\,\,3,\,\,18)\] Alternative \[\therefore \]\[{{A}^{2}}B=diag(2,\,\,-1,\,\,3)diag(2,\,\,-1,\,\,3)\] \[diag(-1,\,\,3,\,\,2)\] \[=diag(4,\,\,1,\,\,9),\,\,diag(-1,\,\,3,\,\,2)\] \[=diag(-4,\,\,3,\,\,18)\]
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