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InterviewSolution
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If the matrix \(\rm A = \left [ \begin{array}{cc} \alpha & \beta \\ \beta & \alpha \end{array}\right ]\) is such that A2 = I, then which one of the following is correct? 1. α = 0, β = 1 OR α = 1, β = 02. α = 1, β = 1 OR α = 1, β = 03. α ≠ 1, β = 04. α = β = 0 |
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Answer» Correct Answer - Option 1 : α = 0, β = 1 OR α = 1, β = 0 Calculation: Here, \(\rm A = \left [ \begin{array}{cc} α & β \\ β & α \end{array}\right ]\) \(\begin{array}{l} \therefore \rm A^{2}=A=\left[\begin{array}{ll} α & β \\ β & α \end{array}\right]\left[\begin{array}{ll} α & β \\ β & α \end{array}\right] \\ \end{array}\) \(=\left[\begin{array}{l} α^{2}+β^{2} & 2 α β\\ 2 α β &α^{2}+β^{2} \end{array}\right] \\ \text { Now } \rm A^{2}=I \\ \) \(\Rightarrow \left[\begin{array}{l} α^{2}+β^{2} & 2 α β \\ 2 α β& α^{2}+β^{2} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ \\ \Rightarrow α^{2}+β^{2}=1, \quad α β=0\) Hence, option (1) is correct. |
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