If z is a complex number and a1,a2,a3,b1,b2,b3  are all real, then =a1z+b1z¯           a2z+b2z¯        a3z+b3z¯b1z+a1z¯             b2z+a2z ¯         b3z+a3z¯b1z+a1           b2z+a2             b3z+a3  is equal to

Using the product of determinants Δ=a1z+b1z¯           a2z+b2z¯        a3z+b3z¯b1z+a1z¯             b2z+a2z ¯         b3z+a3z¯b1z+a1           b2z+a2             b3z+a3  =z  z¯   1z¯  z   11  z  z=a1  b1  0a2  b2  0a3  b3   0=0