All possible values of scalar $ \mathrm{k} $ so that the matrix $ \mathrm{A}^{-1} $ - $ \mathrm{kI} $ is singular where $ \mathrm{A}=\left[\begin{array}{lll}1 & 0 & 2 \ 0 & 2 & 1 \ 1 & 0 & 0\end{array}\right] $

Question

All possible values of scalar $ \mathrm{k} $ so that the matrix $ \mathrm{A}^{-1} $ - $ \mathrm{kI} $ is singular where $ \mathrm{A}=\left[\begin{array}{lll}1 & 0 & 2 \ 0 & 2 & 1 \ 1 & 0 & 0\end{array}\right] $, $ \frac{-1}{2}, 1 $, $ -1, \frac{1}{2} $, $ \frac{1}{2}, \frac{-1}{2} $, $ -1,1 $

MARKS App · Accepted Answer

|A -1 - kI| = 0 |A|A -1 - kI| = 0 (|A| ≠ 0) |I - kA| = 0 I k - A = 0 ⇒ A - 1 k · I = 0 ⇒ |A - λ I| = 0 where λ = 1 k = 1 - λ 0 2 0 2 - λ 1 1 0 - λ = 0 =(1 - λ ) (- λ ) (2 - λ ) + 2(0 - (2 - λ )) = 0 = - λ 3 + 3 λ 2 - 2 λ - 4 + 2 λ = 0 = λ 3 - 3 λ 2 + 4 = 0 ⇒ λ = 2, 2,-1 λ k = - 1 , 1 2

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