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If the rank of the matrix$\begin{bmatrix} \lambda & -1 & 0 \\0 & \lambda & -1 \\-1 & 0 & \lambda \end{bmatrix}$ is $2$, than $\lambda$ is

\[\begin{array}{1 1} (1) 1& (2)2\\ (3) 3& (4) any\; real\; number\end{array}\]

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$A=\begin{bmatrix} \lambda & -1 & 0 \\0 & \lambda & -1 \\-1 & 0 & \lambda \end{bmatrix}$
if $ \lambda=1$
$ A \approx \begin{bmatrix} 1 & -1 & 0 \\0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix}$
$R_3 \to R_3 +R_1$
$ \quad \approx \begin{bmatrix} 1 & -1 & 0 \\0 & 1 & -1 \\ 0 & -1 & 1 \end{bmatrix}$
$R_3 \to R_3 +R_2$
The No. of non zero rows in $A =2$
$p(A)=2$
The value of $\lambda =1$
Hence 1 is the correct answer.
answered May 2, 2014 by meena.p
 

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