Answer Happy

(10 marks) Complex numbers have a representation as two by two matrices with real entries. In this problem you will be asked to show that the matrix operation indeed reproduces the complex number operations. For any complex number z assign a matrix Mz​ as follows: z=a+ib↦Mz​=[ab​−ba​]. For example we have M1+2i​=[12​−21​] or M3i​=[03​−30​]. (a) Show that Mz1​​+Mz2​​=Mz1​+z2​​ and cMz​=Mcz​, for any complex numbers z,z1​,z2​ and real number c. (b) Show that Mz1​​Mz2​​=Mz1​z2​​, for any complex numbers z1​,z2​.
(c) Show that M1/z​=Mz−1​, for any nonzero complex number z. (d) Find an expression for det(Mz​) and MzT​ in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of Mz​ for z=a+ib. (Note that you can only use the techniques used in the section 4.1).