%--------------------------
 % @Author: Jingqiao Hu
 % @Date: 2021-05-18 20:52:12
 % @LastEditTime: 2021-05-18 20:52:13
%--------------------------
% compute the determinat of 3-rank matrix M
function d = det_3rank(M)
    a1 = M(1,:); b1 = M(4,:); c1 = M(7,:);
    a2 = M(2,:); b2 = M(5,:); c2 = M(8,:);
    a3 = M(3,:); b3 = M(6,:); c3 = M(9,:);

    d = a1.*b2.*c3 + b1.*c2.*a3 + c1.*a2.*b3 - a3.*b2.*c1 - b3.*c2.*a1 - c3.*a2.*b1;
end