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Let $A=\left[\begin{array}{cccc}1 & 0 & -1 & -3 \\ 0 & 1 & 1 & k-1 \\ 0 & 0 & k-1 & 1\end{array}\right]$ and $k \in R$. Then, the value of $k$, if exists, for which the rank of $A$ is 2 , is
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The correct answer is:
Does not exist
Given, $A=\left[\begin{array}{cccc}1 & 0 & -1 & -3 \\ 0 & 1 & 1 & k-1 \\ 0 & 0 & k-1 & 1\end{array}\right]$
Applying $R_2 \rightarrow R_2+R_3$,
$$
A=\left[\begin{array}{cccc}
1 & 0 & -1 & -3 \\
0 & 1 & k & k \\
0 & 0 & k-1 & 1
\end{array}\right]
$$
The value of $k$ does not exists as the rank of 2 is not possible in this case.
Applying $R_2 \rightarrow R_2+R_3$,
$$
A=\left[\begin{array}{cccc}
1 & 0 & -1 & -3 \\
0 & 1 & k & k \\
0 & 0 & k-1 & 1
\end{array}\right]
$$
The value of $k$ does not exists as the rank of 2 is not possible in this case.
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