Adjoint of a matrix Just a little help please...
if  c.conj(AB) = conj(B)*conj(A)
and if Transpose(AB) = Transpose(B)*Transpose(A)
then why is Adjoint(AB) = Adjoint(B)*Adjoint(A)
It seems the two operations combined would cancel their effects in reversing order. Thanks
 A: The complex conjugate does not switch the order of matrix multiplication the way you've suggested above. The rules are:
$$ \overline{AB} = \overline{A} \overline{B}, \quad (AB)^T = B^T A^T $$
So since the adjoint for a matrix is complex conjugation + transpose, then we indeed get
$$ (AB)^{\ast} = B^{\ast} A^{\ast}$$
A: You have c.cong() and conj()

But I believe that it is just a typo, conj(AB) = conj(B) conj(A)

The conjugate with matrices can mean element by element, but I imagine that is not the case  here since your formula specifies the order of multiplication. 

It is all to do with whether the elements of interest are scalars (reals or complex) or matrices (commuting or non-commuting elements). The field of reals and complex commute, so no difference need be specified, e.g. ab = ba and $\overline{ab} = \overline{b}\overline{a} = \overline{a}\overline{b}$ and there is no confusion. If the elements however are matrices, then they do not generally commute and the order does matter very much.

I think it is all a matter of what you consider to be the complex conjugate. If one starts with scalars and extends the system (most commonly the reals) to include $\sqrt {-1}$ for example, then the complex and the conjugate do not need to worry about order of multiplication. If then you do that with matrices (with complex values as elements) , or whatever other field, and want a norm that gives a scalar value within the original system (without the extension), and positive only to boot (otherwise how to have the triangle inequality or magnitude comparisons of any sort), then you end up with order and transpose. What then is the complex conjugate? The value (as a function of the input) that when multiplied by the element of interest (input) gives that norm. For matrices that means transpose is necessary. If doing only conjugate of the individual elements, then it is a different field. Is the conjugate on the scalar field or the matrix/vector field? That is the question that needs to be decided beforehand.


Keep It Simple Stupid

If the definition/formula being used requires transpose, do it. If somewhere a contradiction can be proven/found, change definitions/formulas to keep things consistent.
A: As I said in the comments, one way to see that $\overline{AB} =\overline{B}\overline{A}$ can't possibly work comes from taking $A$ to be $2\times 2$ and $B$ to be $2\times 1$.  Then you can't swap the order and still multiply.
I suspect the reason wikipedia got it wrong is because of the following.  For complex (and real) numbers, we have $\overline{ab} = \overline{a}\overline{b} = \overline{b}\overline{a}$ since the order doesn't matter.
If one tries to generalize this to the quaternions (where order of multiplication does matter), then the correct rule is $\overline{ab} = \overline{b}\overline{a}$.  From here, it's easy to guess that non commutative things swap when taking complex conjugates and then apply this reasoning to matrices (where it is wrong).
