Question

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

• A. $2abc$
• B. $8abc$
• C. $4a^2{b}^2{c}^2$
• D. $2a^2{b}^2{c}^2$

Answer 1

Answer: (C)

$4a^2{b}^2{c}^2$

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

$=a^2(b^2{c}^2-ac(b^2+bc))-bc(c^2(a^2+ab)-a^{2}bc)+(ac+c^2)((a^2+ab)(b^2+bc)-{ab}^3)$

$=a^2{b}^2{c}^2-a^3{b}^{2}c-a^3{bc}^2-bc(a^2{c}^2-{abc}^2-a^{2}bc)+(ac+c^2)(a^3{b}^2+a^{2}bc+{ab}^2+{ab}^{2}c-{ab}^3)$

$=a^2{b}^2{c}^2-a^3{b}^{2}c-a^3{bc}^2-a^2{bc}^3+{ab}^2{c}^3+a^2{b}^2{c}^2+a^2{b}^{2}c+a^3{bc}^2+a^2{b}^2{c}^2+a^2{b}^2{c}^2+a^2{bc}^3+{ab}^2{c}^2$

$=4a^2{b}^2{c}^2$