Evaluate the following-x 111 x 111 x

Let nbsp; #8710;=x111x111x. #8710;=x111x111x #160; #160; #160;=x 11 x01x101 xx 1 #160; #160; #160; #160; #160; #160;Applying #160;R1 ...

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Standard XII

Mathematics

Factorization

Evaluate the ...

Question

Evaluate the following:

$|\begin{matrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{matrix}|$

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Solution

Let $∆ = |\begin{matrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{matrix}|$ .
$∆ = |\begin{matrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{matrix}| = |\begin{matrix} x - 1 & 1 - x & 0 \\ 1 & x & 1 \\ 0 & 1 - x & x - 1 \end{matrix}| [Applying R_{1} \to R_{1} - R_{2} and R_{3} \to R_{3} - R_{2}] = {(x - 1)}^{2} |\begin{matrix} 1 & - 1 & 0 \\ 1 & x & 1 \\ 0 & - 1 & 1 \end{matrix}| = {(x - 1)}^{2} |\begin{matrix} 1 & - 1 & 0 \\ 1 & x + 1 & 1 \\ 0 & 0 & 1 \end{matrix}| [Applying C_{2} \to C_{2} + C_{3}] = {(x - 1)}^{2} (x + 1 + 1) [Expanding along last row] = {(x - 1)}^{2} (x + 2) ∴ ∆ = {(x - 1)}^{2} (x + 2)$

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Evaluate the following: x111x111x

Let ∆=x111x111x. ∆=x111x111x =x-11-x01x101-xx-1 Applying R1→R1-R2 and R3→R3-R2 =x-121-101x10-11 =x-121-101x+11001 Applying C2→C2+C3 =x-12(x+1+1) Expanding along last row =x-12(x+2)∴ ∆=x-12(x+2)

Evaluate the following:

$|\begin{matrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{matrix}|$