Question

$\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ac& bc& c^2+1\end{array}\right|=f(a,b,c)$, $f$ is

• A. Non-Homogeneous of degree 2
• B. Homogeneous of degree 4
• C. Homogeneous of degree 6
• D. Non-Homogeneous of degree 6

Answer 1

The correct option is A Non-Homogeneous of degree 2

$L.H.S=\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ac& bc& c^2+1\end{array}\right|$

Multiplying $C_1,C_2,C_3$ by a, b, c respectively

$=\frac{1}{abc}\left|\begin{array}{ccc}a(a^2+1)& a{b}^2& a{c}^2\\ a^{2}b& b(b^2+1)& b{c}^2\\ a^{2}c& b^{2}c& c(c^2+1)\end{array}\right|$

Now taking common a, b, c from $R_1,R_2,R_3$ respectively

$=\frac{abc}{abc}\left|\begin{array}{ccc}{a}^2+1& b^2& c^2\\ a^2& b^2+1& c^2\\ a^2& b^2& c^2+1\end{array}\right|$

$=\left|\begin{array}{ccc}1+a^2+b^2+c^2& b^2& c^2\\ 1+a^2+b^2+c^2& b^2+1& c^2\\ 1+a^2+b^2+c^2& b^2& c^2+1\end{array}\right|[C_1\to C_1+C_2+C_3]$

$=(1+a^2+b^2+c^2)\left|\begin{array}{ccc}1& b^2& c^2\\ 1& b^2+1& c^2\\ 1& b^2& c^2+1\end{array}\right|$

$=(1+a^2+b^2+c^2)\left|\begin{array}{ccc}1& b^2& c^2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|[R_2\to R_2-R_1$ and $R_3\to R_3-R_1]$

$=(1+a^2+b^2+c^2)$ $\left(\mathrm{1.1.1}\right)$ (expanding along first column)

$=1+a^2+b^2+c^2$ which is non-homogeneous with degree $2$.