Let P be a $2 \times 2$ matrix such that $P [_{- 1}^{1}] = [_{2}^{- 1}]$ and $P^{2} [_{- 1}^{1}] = [_{0}^{1}]$ . If $x_{1} a n d x_{2}$ are two values of x for which $| P - x I | = 0$ where I is an identity matrix of order 2 then the value of $\frac{x_{1}^{2} + x_{2}^{2}}{2} =$

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Detailed Solution

$L e t P = [\begin{matrix} a & b \\ c & d \end{matrix}] T h e n P [\begin{matrix} 1 \\ - 1 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \end{matrix}] \Rightarrow a - b = - 1 c - d = 2 P^{2} [\begin{matrix} 1 \\ - 1 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}] \Rightarrow P . P [\begin{matrix} 1 \\ - 1 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}] \Rightarrow [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} - 1 \\ 2 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}] \Rightarrow - a + 2 b = 1, - c + 2 d = 0 a = - 1, b = 0, c = 4, d = 2 | \begin{matrix} - 1 - x & 0 \\ 4 & 2 - x \end{matrix} | = 0 \Rightarrow (x + 1) (x - 2) = 0$