Question

Prove that $\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ca& cb& c^2+1\end{array}\right|=1+a^2+b^2+c^2$

Answer 1

Let , L.H.S.

Taking out common $a,b,c$

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

multiplying $a,b,c$ in $c_1,c_2,c_3$ respectively

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

Applying $c_1\to c_1+c_2+c_3$

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

Taking out common $1+a^2+b^2+c^2$ from $c_1$

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

Applying

$R_2\to R_2-R_1$

$R_3\to R_3-R_1$

$=\left(1+a^2+b^2+c^2\right)\left|\begin{array}{ccc}1& b^2& c^2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$

Expanding along $c_1$

$=1+a^2+b^2+c^2$