We consider complex matrices, that is, matrices with complex entries. If A = (aij) is such...

We consider complex matrices, that is, matrices with complex entries. If A = (aij) is such a matrix, we denote by the conjugate of A, that is, the matrix , where z is the conjugate of the complex number z (for example, if z = 3 + 5i, then z = 3 – 5i).

Prove the following:

a) If z1 and z2 are complex numbers, then

b) If A and B are matrices and c is a complex scalar, then

c) Every complex matrix A can be written uniquely as A1 + iA2, where A1 and A2 are real matrices, and We call A1 the real part of A, A2 the imaginary par of A.

d) If A is a square matrix, then and, if A is nonsingular, then

e) A is a real matrix if and only if A =