Question

Using properties of determination, 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

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

$\therefore a^2+1(b^2+1\times c^2+1-bc\times cb)-ab(ab\times c^2+1-bc\times ca)+ac(ab\times cb-b^2+1\times ca)$

$\therefore a^2+1(b^2/c^2+b^2+c^2+1-b^2/c^2)-ab(ab{c}^2+ab-ab{c}^2)+ac(a{b}^{2}c-a{b}^{2}c+ca)$

$=a^2+1(b^2+c^2+1)-ab\left(ab\right)+ac\left(ca\right)$

$=a^2{b}^2-a^2{c}^2+a^2+b^2+c^2+1-a^2{b}^2+a^2{c}^2$

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