Question

Solve the equation:
$\left|\begin{array}{ccc}u+a^{2}x& w^{\prime }+abx& v^{\prime }+acx\\ w^{\prime }+abx& v+b^{2}x& u^{\prime }+bcx\\ v^{\prime }+acx& u^{\prime }+bcx& w+c^{2}x\end{array}\right|=0$,
expressing the result by means of determinants.

Answer 1

The determinant can be expressed as a sum of eight determinants

The terms containing $x^3$ will be;-

$\left|\begin{array}{ccc}{a}^{2}x& abx& acx\\ abx& b^{2}x& bcx\\ acx& bcx& c^{2}x\end{array}\right|$ or $abc{x}^3\left|\begin{array}{ccc}a& a& a\\ b& b& b\\ c& c& c\end{array}\right|$

therefore co efficient of $x^3$ will be $0$

The terms containing $x^2$ will be obtained from

$\left|\begin{array}{ccc}{u}^{{}^{\prime }}& abx& acx\\ w^{{}^{\prime }}& b^{2}x& bcx\\ v^{{}^{\prime }}& bcx& c^{2}x\end{array}\right|+\left|\begin{array}{ccc}{a}^{2}x& w^{{}^{\prime }}& acx\\ abc& v& bcx\\ acx& u^{{}^{\prime }}& c^{2}c\end{array}\right|+\left|\begin{array}{ccc}{a}^{2}x& abx& v^{{}^{\prime }}\\ abx& b^{2}x& u^{{}^{\prime }}\\ acx& bcx& w\end{array}\right|$

By taking $bc{x}^2$ from first determinant second and third column becomes same

Hence, the co efficient of $x^2$ is also zero

The co efficient of x

$=a\left|\begin{array}{ccc}a& w^{{}^{\prime }}& v^{{}^{\prime }}\\ b& v& u^{{}^{\prime }}\\ c& u^{{}^{\prime }}& w\end{array}\right|+b\left|\begin{array}{ccc}u& a& v^{{}^{\prime }}\\ w^{{}^{\prime }}& b& u^{{}^{\prime }}\\ v^{{}^{\prime }}& c& w\end{array}\right|+c\left|\begin{array}{ccc}u& w^{{}^{\prime }}& a\\ w^{{}^{\prime }}& v& b\\ v^{{}^{\prime }}& u^{{}^{\prime }}& c\end{array}\right|$

$=-\left|\begin{array}{cccc}0& a& b& c\\ a& u& w^{{}^{\prime }}& u^{{}^{\prime }}\\ b& w^{{}^{\prime }}& v& v^{{}^{\prime }}\\ c& v^{{}^{\prime }}& u^{{}^{\prime }}& w\end{array}\right|$

Lastly, the term independent of x is

$\left|\begin{array}{ccc}u& w^{{}^{\prime }}& v^{{}^{\prime }}\\ w^{{}^{\prime }}& v& u^{{}^{\prime }}\\ v^{{}^{\prime }}& u^{{}^{\prime }}& w\end{array}\right|$

$x=\left|\begin{array}{ccc}u& w^{{}^{\prime }}& v^{{}^{\prime }}\\ w^{{}^{\prime }}& v& u^{{}^{\prime }}\\ v^{{}^{\prime }}& u^{{}^{\prime }}& w\end{array}\right|÷\left|\begin{array}{cccc}u& w^{{}^{\prime }}& v^{{}^{\prime }}& a\\ w^{{}^{\prime }}& v& u^{{}^{\prime }}& b\\ v^{{}^{\prime }}& u^{{}^{\prime }}& w& c\\ a& b& c& 0\end{array}\right|$