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

Answer 1

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

$C_1\to C_1+\frac{y}{x}{C}_2$
$\left|\begin{array}{ccc}0& -xy& -x^2\\ a+\frac{yb}{x}& b& c\\ a^{\prime }+\frac{y{b}^{\prime }}{x}& b^{\prime }& c^{\prime }\end{array}\right|$
$C_3\to C_3+\frac{x}{y}{C}_2$
$\left|\begin{array}{ccc}0& -xy& 0\\ a+\frac{yb}{x}& b& c+\frac{xb}{y}\\ a^{\prime }+\frac{y{b}^{\prime }}{x}& b^{\prime }& c^{\prime }+\frac{xb}{y}\end{array}\right|$
$+xy\left|\begin{array}{cc}a+\frac{yb}{x}& c+a+\frac{xb}{x}\\ a^{\prime }+\frac{y{b}^{\prime }}{x}& c^{\prime }+\frac{x{b}^{\prime }}{x}\end{array}\right|$
$\left|\begin{array}{cc}axy+y^{2}b& cxy+b{x}^2\\ a^{\prime }xy+y^2{b}^{\prime }& c^{\prime }xy+b^{\prime }{x}^2\end{array}\right|$