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જો $a,b,c$ એ ધન પૂર્ણાંક હોય , તો $\Delta = \left| {\,\begin{array}{*{20}{c}}{{a^2} + x}&{ab}&{ac}\\{ab}&{{b^2} + x}&{bc}\\{ac}&{bc}&{{c^2} + x}\end{array}\,} \right|$ એ . . . વડે વિભાજ્ય છે.
${x^3}$
${x^2}$
$({a^2} + {b^2} + {c^2})$
એકપણ નહી.
Solution
(b) $\Delta = \frac{1}{{abc}}\,\left| {\,\begin{array}{*{20}{c}}{{a^3} + ax}&{{a^2}b}&{{a^2}c}\\{a{b^2}}&{{b^3} + bx}&{{b^2}c}\\{{c^2}a}&{{c^2}b}&{{c^3} + cx}\end{array}\,} \right|$
=$({{a}^{2}}+{{b}^{2}}+{{c}^{2}}+x)\times \left| \,\begin{matrix}
1 & 1 & 1 \\
{{b}^{2}} & {{b}^{2}}+x & {{b}^{2}} \\
{{c}^{2}} & {{c}^{2}} & {{c}^{2}}+x \\
\end{matrix}\, \right|$
$\{$Applying ${R_1} \to {R_1} + {R_2} + {R_3}\} $
$ = ({a^2} + {b^2} + {c^2} + x)\,\left| {\,\begin{array}{*{20}{c}}1&0&0\\{{b^2}}&x&0\\{{c^2}}&0&x\end{array}\,} \right|$$\left. \begin{array}{l}{C_2} \to {C_2} – {C_1}\\{C_3} \to {C_3} – {C_1}\end{array} \right\}$
= ${x^2}({a^2} + {b^2} + {c^2} + x)$.
Hence $\Delta $ is divisible by ${x^2}$ as well as by $ x.$
