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If $A=\left[\begin{array}{cc}2-k & 2 \\ 1 & 3-k\end{array}\right]$ is a singular matrix, then the value of $5 k-k^2$ is
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Given, $A=\left[\begin{array}{cc}2-k & 2 \\ 1 & 3-k\end{array}\right]$
Since, the Matrix $A$ is singular
$$
\begin{aligned}
& \therefore \quad|A|=0 \\
& \Rightarrow \quad\left|\begin{array}{cc}
2-k & 2 \\
1 & 3-k
\end{array}\right|=0 \\
& \Rightarrow \quad(2-k)(3-k)-2=0 \\
& \Rightarrow \quad 6-5 k+k^2-2=0 \\
& \Rightarrow \quad 4-5 k+k^2=0 \\
& \Rightarrow \quad 5 k-k^2=4 \\
&
\end{aligned}
$$
Since, the Matrix $A$ is singular
$$
\begin{aligned}
& \therefore \quad|A|=0 \\
& \Rightarrow \quad\left|\begin{array}{cc}
2-k & 2 \\
1 & 3-k
\end{array}\right|=0 \\
& \Rightarrow \quad(2-k)(3-k)-2=0 \\
& \Rightarrow \quad 6-5 k+k^2-2=0 \\
& \Rightarrow \quad 4-5 k+k^2=0 \\
& \Rightarrow \quad 5 k-k^2=4 \\
&
\end{aligned}
$$
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