A) \[\alpha ={{a}^{2}}+{{b}^{2}},\,\beta =ab\]
B) \[\alpha ={{a}^{2}}+{{b}^{2}},\,\beta =2ab\]
C) \[\alpha ={{a}^{2}}+{{b}^{2}},\,\beta ={{a}^{2}}-{{b}^{2}}\]
D) \[\alpha =2ab,\,\,\beta ={{a}^{2}}+{{b}^{2}}\]
Correct Answer: B
Solution :
Using properties of matrices \[{{A}^{2}}=A\]. A and compare it with given value of \[{{A}^{2}}\]. \[A=\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right],{{A}^{2}}=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right]\] \[{{A}^{2}}=A\,.\,A=\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right]\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right]\] \[\Rightarrow \] \[{{A}^{2}}=\left[ \begin{matrix} {{a}^{2}}+{{b}^{2}} & ab+ba \\ ba+ab & {{b}^{2}}+{{a}^{2}} \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} {{a}^{2}}+{{b}^{2}} & 2ab \\ 2ab & {{a}^{2}}+{{b}^{2}} \\ \end{matrix} \right]\] \[\Rightarrow \] \[\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right]=\left[ \begin{matrix} {{a}^{2}}+{{b}^{2}} & 2\,ab \\ 2\,ab & {{a}^{2}}+{{b}^{2}} \\ \end{matrix} \right]\] \[\Rightarrow \] \[\alpha ={{a}^{2}}+{{b}^{2}}\], \[\beta =2\,ab\]
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