====== 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.
=====Question 12=====
Find the value of $\lambda $, if $A$ is singular matrix, where $A=\begin{bmatrix}-\lambda  & 1 & 0  \\1 & -\lambda  & 1  \\0 & 1 & -\lambda \end{bmatrix}$
====Solution====
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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