Question

Using properties of determinants, prove the following:
$\left|\begin{array}{ccc}{a}^2& bc& ac+c^2\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|=4a^2{b}^2{c}^2$.

Answer 1

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

Adding row $2$ and row $3$ to row $1$ and taking out common factor, we get:

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

Subtracting row $3$ from row $1,$ we get:

$2\left|\begin{array}{ccc}{a}^2& 0& ac\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|$

Subtracting row $1$ from row $2,$ we get:

$2\left|\begin{array}{ccc}{a}^2& 0& ac\\ ab& b^2& 0\\ ab& b^2+bc& c^2\end{array}\right|$

Subtracting row $2$ from row $3,$ we get:

$2\left|\begin{array}{ccc}{a}^2& 0& ac\\ ab& b^2& 0\\ 0& bc& c^2\end{array}\right|$

the above determinant on expansion gives:

$2a^2{b}^2{c}^2+2ac.ab.bc=4a^2{b}^2{c}^2.$