The values of x for which the value of A-0- where A- x-1 1 1- 1 x-1 1- 1 1 x-1 -

|A| = x 1 amp; 1 amp; 1 nbsp; 1 amp; x 1 amp; 1 1 amp;1 amp; x 1 Expanding along the first row |A| = (x 1) nbsp;x 1 amp; 1 nbsp; ...

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Standard XII

Mathematics

Chain Rule of Differentiation

The values of...

Question

The value(s) of $x$ for which the value of $det A = 0,$ where $A = ⎡ ⎢ ⎣ \begin{matrix} x - 1 & 1 & 1 1 & x - 1 & 1 1 & 1 & x - 1 \end{matrix} ⎤ ⎥ ⎦$

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Solution

The correct option is D $2$
$| A | = ∣ ∣ ∣ ∣ \begin{matrix} x - 1 & 1 & 1 1 & x - 1 & 1 1 & 1 & x - 1 \end{matrix} ∣ ∣ ∣ ∣$
Expanding along the first row
$| A | = (x - 1) ∣ ∣ ∣ \begin{matrix} x - 1 & 1 1 & x - 1 \end{matrix} ∣ ∣ ∣ - 1 ∣ ∣ ∣ \begin{matrix} 1 & 1 1 & x - 1 \end{matrix} ∣ ∣ ∣ + 1 ∣ ∣ ∣ \begin{matrix} 1 & x - 1 1 & 1 \end{matrix} ∣ ∣ ∣$
$= (x - 1) ((x - 1) (x - 1) - 1) - 1 ((x - 1) - 1) + 1 (1 - (x - 1))$
$= (x - 1) (x^{2} - 2 x + 1 - 1) - (x - 2) + (2 - x)$
$= x (x - 1) (x - 2) - (x - 2) - (x - 2)$
$= (x - 2)$ [ $x (x - 1) - 2$ ]
$= (x - 2) (x^{2} - x - 2)$
Now $| A | = 0$
$\Rightarrow$ $(x - 2) (x^{2} - x - 2) = 0$
$\Rightarrow$ $(x - 2) (x - 2) (x + 1) = 0$
$∴ x = - 1$ or $2$

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Chain Rule of Differentiation

Standard XII Mathematics

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The value(s) of x for which the value of detA=0, where A=⎡⎢⎣x−1111x−1111x−1⎤⎥⎦

The value(s) of $x$ for which the value of $det A = 0,$ where $A = ⎡ ⎢ ⎣ \begin{matrix} x - 1 & 1 & 1 1 & x - 1 & 1 1 & 1 & x - 1 \end{matrix} ⎤ ⎥ ⎦$