Let M be a 2 × 2 matrix such that M1−1=−12 and M21−1=10 If x1 and x2 (x1 > x2) are the twovalues of x for which det (M–XI)=0 where I is identity matrix of order 2 then the value of 5x1 + 2x2 is

Let m=a bc d∴a−b=−1,c−d=2⇒−a+2b=1,−c+2d=0⇒m=−1042∴|m−xI|=0−1-x042-x=0⇒(x−2)(x+1)=0⇒x1=+2x2=−1⇒5x1+2x2

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Let M be a 2 × 2 matrix such that $M [\begin{matrix} 1 \\ - 1 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \end{matrix}] and M^{2} [\begin{matrix} 1 \\ - 1 \end{matrix}] = [\begin{array}{l} 1 \\ 0 \end{array}]$ If x₁ and x₂ (x₁ > x₂) are the two
values of x for which det (M–XI)=0 where I is identity matrix of order 2 then the value of 5x₁ + 2x₂ is

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Detailed Solution

Let $m = [\begin{array}{ll} a b \\ c d \end{array}]$
$\begin{array}{l} ∴ a - b = - 1, c - d = 2 \\ \Rightarrow - a + 2 b = 1, - c + 2 d = 0 \Rightarrow m = [\begin{array}{cc} - 1 & 0 \\ 4 & 2 \end{array}] \\ ∴ | m - xI | = 0 \\ |\begin{array}{cc} - 1 - x & 0 \\ 4 & 2 - x \end{array}| = 0 \\ \Rightarrow (x - 2) (x + 1) = 0 \\ \Rightarrow x_{1} = + 2 x_{2} = - 1 \Rightarrow 5 x_{1} + 2 x_{2} \end{array}$