First slide

Applications of determinant

Question

For what value of $x,$ the matrix $[\begin{matrix} 3 - x & 2 & 2 \\ 2 & 4 - x & 1 \\ - 2 & - 4 & - 1 - x \end{matrix}]$ is singular

Easy

A

$x = 1$
B

$x = 2$
C

$x = 0$
D

$x = 3$

Solution

Since, the given matrix is singular
$\Rightarrow |\begin{matrix} 3 - x & 2 & 2 \\ 2 & 4 - x & 1 \\ - 2 & - 4 & - 1 - x \end{matrix}| = 0$
Applying $R_{2} \to R_{2} + R_{3}$
$\Rightarrow |\begin{matrix} 3 - x & 2 & 2 \\ 2 & - x & - x \\ - 2 & - 4 & - 1 - x \end{matrix}| = 0 \Rightarrow x |\begin{matrix} 3 - x & 2 & 2 \\ 0 & 1 & 1 \\ 2 & 4 & 1 + x \end{matrix}| = 0$
$\Rightarrow x {(3 - x) (1 + x - 4) - 0 + 2 (2 - 2)} = 0$
$\Rightarrow x {(3 - x) (x - 3)} = 0 \Rightarrow x = 0, 3$

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