The determinant left|begin{array}{ccc}a^{2}+1 | vedclass

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(B) Let $\Delta = \left|\begin{array}{ccc}a^{2}+10 & a b & a c \ a b & b^{2}+10 & b c \ a c & b c & c^{2}+10\end{array}ight|$.Multiply $R_1$ by $a

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divisible by $100$

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(B) Let $\Delta = \left|\begin{array}{ccc}a^{2}+10 & a b & a c \ a b & b^{2}+10 & b c \ a c & b c & c^{2}+10\end{array}ight|$.Multiply $R_1$ by $a$,$R_2$ by $b$,and $R_3$ by $c$:$\Delta = \frac{1}{abc} \left|\begin{array}{ccc}a(a^{2}+10) & a^2 b & a^2 c \ ab^2 & b(b^{2}+10) & b^2 c \ ac^2 & bc^2 & c(c^{2}+10)\end{array}ight|$.Taking $a, b, c$ common from $C_1, C_2, C_3$ respectively:$\Delta = \frac{abc}{abc} \left|\begin{array}{ccc}a^{2}+10 & a^{2} & a^{2} \ b^{2} & b^{2}+10 & b^{2} \ c^{2} & c^{2} & c^{2}+10\end{array}ight|$.Applying $R_1 ightarrow R_1 + R_2 + R_3$:$\Delta = (a^2+b^2+c^2+10) \left|\begin{array}{ccc}1 & 1 & 1 \ b^2 & b^2+10 & b^2 \ c^2 & c^2 & c^2+10\end{array}ight|$.Applying $C_2 ightarrow C_2 - C_1$ and $C_3 ightarrow C_3 - C_1$:$\Delta = (a^2+b^2+c^2+10) \left|\begin{array}{ccc}1 & 0 & 0 \ b^2 & 10 & 0 \ c^2 & 0 & 10\end{array}ight|$.Expanding along $R_1$:$\Delta = (a^2+b^2+c^2+10) 	imes (10 	imes 10) = 100(a^2+b^2+c^2+10)$.Thus,the determinant is divisible by $100$.

The determinant $\left|\begin{array}{ccc}a^{2}+10 & a b & a c \\ a b & b^{2}+10 & b c \\ a c & b c & c^{2}+10\end{array}\right|$ is

Similar Questions

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