First slide

Elementary row operations (Transformation)

Question

If the rank of $[\begin{matrix} x & x & x \\ x & x^{2} & x \\ x & x & x + 1 \end{matrix}]$ is 1, then

Easy

A

$x = 0 (o r) x = 1$
B

$x = 0$
C

$x = 1$
D

$x = \pm 2$

Solution

$[\begin{matrix} x & x & x \\ x & x^{2} & x \\ x & x & x + 1 \end{matrix}]$ $\begin{array}{l} R_{2} - R_{1}, \\ R_{3} - R_{1} \end{array}$
$~ [\begin{matrix} x & x & x \\ 0 & x^{2} - x & 0 \\ 0 & 0 & 1 \end{matrix}]$
If rank =1, Then $x^{2} - x = 0$
$\Rightarrow x (x - 1) = 0$
$\Rightarrow x = 0 (∵ i f x = 1, t h e n r a n k = 2)$
Is the possible answer.

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