Question

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
(A) divisible by 10 but not by 100
(B) divisible by 100
(C) not divisible by 100
(D) not divisible by 10

Answer 1

Ans: (B)
Hint \(:=\frac{1}{a b c}\left|\begin{array}{ccc}a\left(a^{2}+10\right) & a^{2} b & a^{2} c \\ a b^{2} & b\left(b^{2}+10\right) & b^{2} c \\ a c^{2} & b c^{2} & c\left(c^{2}+10\right)\end{array}\right|=\frac{a b c}{a b c}\left|\begin{array}{cccc}a^{2}+10 & a^{2} & a^{2} \\ b^{2} & b^{2}+10 & b^{2} \\ c^{2} & c^{2} & c^{2}+10\end{array}\right|\) \(\mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}\)
$$
=\left(a^{2}+b^{2}+c^{2}+10\right)\left|\begin{array}{ccc}
1 & 1 & 1 \\
b^{2} & b^{2}+10 & b^{2} \\
c^{2} & c^{2} & c^{2}+10
\end{array}\right|
$$
$$
\begin{aligned}
&=\left(a^{2}+b^{2}+c^{2}+10\right)\left|\begin{array}{ccc}
1 & 0 & 0 \\
b^{2} & 10 & 0 \\
c^{2} & 0 & 10
\end{array}\right|\left[\begin{array}{l}
c_{2}-c_{1} \\
c_{3}-c 1
\end{array}\right] \\
&=\left(a^{2}+b^{2}+c^{2}+10\right) 100
\end{aligned}
$$