Question

Calculate the values of the determinants:
$\left|\begin{array}{cccc}0& x& y& z\\ -x& 0& c& b\\ -y& -c& 0& a\\ -z& -b& -a& 0\end{array}\right|$.

Answer 1

$\left|\begin{array}{cccc}0& x& y& z\\ -x& 0& c& b\\ -y& -c& 0& a\\ -z& -b& -a& 0\end{array}\right|$

$=0\times \left|\begin{array}{ccc}0& c& b\\ -c& 0& a\\ -b& -a& 0\end{array}\right|–x\times \left|\begin{array}{ccc}-x& c& b\\ -y& 0& a\\ -z& -a& 0\end{array}\right|+y\times \left|\begin{array}{ccc}-x& 0& b\\ -y& -c& a\\ -z& -b& 0\end{array}\right|–z\times \left|\begin{array}{ccc}-x& 0& c\\ -y& -c& 0\\ -z& -b& -a\end{array}\right|$

$=-x(-a^x-acz+abz)+y(-abx+b^{2}y-bcz)-z(-acx+bcy-c^{2}z)$

$=a^2{x}^2+b^2{y}^2+c^2{z}^2+2acxz-2abxy-2bcyz$