If a, b, c are positive integers,then the det | vedclass

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(B) To evaluate the determinant $\Delta = \begin{vmatrix} a^2 + x & ab & ac \ ab & b^2 + x & bc \ ac & bc & c^2 + x \end{vmatrix}$,we multiply $R_1$

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$x^2$

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(B) To evaluate the determinant $\Delta = \begin{vmatrix} a^2 + x & ab & ac \ ab & b^2 + x & bc \ ac & bc & c^2 + x \end{vmatrix}$,we multiply $R_1$ by $a$,$R_2$ by $b$,and $R_3$ by $c$ and divide the determinant by $abc$: $\Delta = \frac{1}{abc} \begin{vmatrix} a(a^2 + x) & a^2b & a^2c \ ab^2 & b(b^2 + x) & b^2c \ ac^2 & bc^2 & c(c^2 + x) \end{vmatrix}$ Taking $a, b, c$ common from $C_1, C_2, C_3$ respectively: $\Delta = \frac{abc}{abc} \begin{vmatrix} a^2 + x & a^2 & a^2 \ b^2 & b^2 + x & b^2 \ c^2 & c^2 & c^2 + x \end{vmatrix} = \begin{vmatrix} a^2 + x & a^2 & a^2 \ b^2 & b^2 + x & b^2 \ c^2 & c^2 & c^2 + x \end{vmatrix}$ Applying $R_1 	o R_1 + R_2 + R_3$: $\Delta = \begin{vmatrix} a^2 + b^2 + c^2 + x & a^2 + b^2 + c^2 + x & a^2 + b^2 + c^2 + x \ b^2 & b^2 + x & b^2 \ c^2 & c^2 & c^2 + x \end{vmatrix}$ Taking $(a^2 + b^2 + c^2 + x)$ common from $R_1$: $\Delta = (a^2 + b^2 + c^2 + x) \begin{vmatrix} 1 & 1 & 1 \ b^2 & b^2 + x & b^2 \ c^2 & c^2 & c^2 + x \end{vmatrix}$ Applying $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_1$: $\Delta = (a^2 + b^2 + c^2 + x) \begin{vmatrix} 1 & 0 & 0 \ b^2 & x & 0 \ c^2 & 0 & x \end{vmatrix} = (a^2 + b^2 + c^2 + x)(x^2)$ Thus,$\Delta$ is divisible by $x^2$.

If $a, b, c$ are positive integers,then the determinant $\Delta = \begin{vmatrix} a^2 + x & ab & ac \\ ab & b^2 + x & bc \\ ac & bc & c^2 + x \end{vmatrix}$ is divisible by

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