Question

If $a,b,c$ are non negative real numbers and $\left|\begin{array}{ccc}(a^2+x^2)& ab& ac\\ ab& (b^2+x^2)& bc\\ ac& bc& (c^2+x^2)\end{array}\right|$ is divisible by $x^n$, where $n\in N$, then the maximum possible value of $n$ is

Answer 1

Let $A=\left|\begin{array}{ccc}(a^2+x^2)& ab& ac\\ ab& (b^2+x^2)& bc\\ ac& bc& (c^2+x^2)\end{array}\right|$
Using row operations, we get
$R_1\to a{R}_1,R_2\to b{R}_2,R_3\to c{R}_3\Rightarrow A=\frac{1}{abc}\left|\begin{array}{ccc}a(a^2+x^2)& a^{2}b& a^{2}c\\ a{b}^2& b(b^2+x^2)& b^{2}c\\ a{c}^2& b{c}^2& c(c^2+x^2)\end{array}\right|\Rightarrow A=\left|\begin{array}{ccc}(a^2+x^2)& a^2& a^2\\ b^2& (b^2+x^2)& b^2\\ c^2& c^2& (c^2+x^2)\end{array}\right|$

$R_1\to R_1+R_2+R_3\Rightarrow A=(a^2+b^2+c^2+x^2)\left|\begin{array}{ccc}1& 1& 1\\ b^2& b^2+x^2& b^2\\ c^2& c^2& (c^2+x^2)\end{array}\right|$

$C_2\to C_2-C_1,C_3\to C_3-C_1\Rightarrow A=(a^2+b^2+c^2+x^2)\left|\begin{array}{ccc}1& 0& 0\\ b^2& x^2& 0\\ c^2& 0& x^2\end{array}\right|\Rightarrow A=x^4(a^2+b^2+c^2+x^2)$

Hence, the maximum value of $n=4$