Question

The determinant $\left|\begin{array}{ccc}{y}^2& -xy& x^2\\ a& b& c\\ a^{\prime }& b^{\prime }& c^{\prime }\end{array}\right|$ is equivalent to

• A. $\left|\begin{array}{cc}bx+ay& cx+by\\ b^{\prime }x+a^{\prime }y& c^{\prime }x+b^{\prime }y\end{array}\right|$
• B. $\left|\begin{array}{cc}ax-by& bx+cy\\ -a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$
• C. $\left|\begin{array}{cc}bx+cy& ax+by\\ b^{\prime }x+c^{\prime }y& a^{\prime }x+b^{\prime }y\end{array}\right|$
• D. $\left|\begin{array}{cc}ax+by& bx+cy\\ a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$

Answer 1

Answer: (D)

$\left|\begin{array}{cc}ax+by& bx+cy\\ a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$

$\left|\begin{array}{ccc}{y}^2& -xy& x^2\\ a& b& c\\ a^{\prime }& b^{\prime }& c^{\prime }\end{array}\right|=y^2(b{c}^{\prime }-b^{\prime }c)+xy(a{b}^{\prime }-a^{\prime }b)+x^2(a{c}^{\prime }-a^{\prime }c)=y^{2}b{c}^{\prime }-y^2{b}^{\prime }c+xya{b}^{\prime }-xy{a}^{\prime }b+x^{2}a{c}^{\prime }-x^2{a}^{\prime }c=(ax+by)(b^{\prime }x+c^{\prime }y)-(a^{\prime }x+b^{\prime }y)(bx+cy)=\left|\begin{array}{cc}ax+by& bx+cy\\ a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$