Question 12, Exercise 2.2

Solutions of Question 12 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Find the value of $\lambda $, if $A$ is singular matrix, where $A=\begin{bmatrix}-\lambda & 1 & 0 \\1 & -\lambda & 1 \\0 & 1 & -\lambda \end{bmatrix}$

Given $$A=\left[ \begin{matrix} -\lambda & 1 & 0 \\ 1 & -\lambda & 1 \\ 0 & 1 & -\lambda \\ \end{matrix} \right]$$ $$|A|=-\lambda (\lambda ^2-1)-1(-\lambda -0)+0$$ $$=-\lambda (\lambda ^2-1)+\lambda $$ $$|A|=\lambda (1-(\lambda ^2-1))$$ $A$ is singular. $$\Rightarrow |A|=0$$ $$\lambda (1-(\lambda ^2-1))=0$$ $$\lambda =0$$ $$1-(\lambda ^2-1)=0$$ $$\lambda ^2-1=1$$ $$\lambda ^2=2$$ $$\lambda =\pm \sqrt{2}$$ $$\lambda =0,\pm \sqrt{2}$$

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