Question

Find the value of the determinant without expansion
$\left|\begin{array}{ccc}{b}^2-ab& b-c& bc-ac\\ ab-a^2& a-b& b^2-ab\\ bc-ac& c-a& ab-a^2\end{array}\right|$

Answer 1

Taking out (b-a) from $C_1$ and $C_3$, we get

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

Now operating $C_1-C_3$ we get $C_1$ and $C_2$ identical in the newly formed determinant.

Hence $△=0.$