To do:

Try concrete calculations with transposes coupled wth inverses.

Practice the reduction of symmetric matrices.

Prove that every square matrix can be written as where is symmetric and is antisymmetric. Hint is to take combinations of .

Study this formal proof why equals . First of all, let us agree that denotes the entry in the -th row and -th column of . Similar notation holds for as well as their transposes. Then by definition, we have:

and the formal product formula says that
, where is the common number of columns of and rows of .


So, we see that the -th entry of is which equals
and this easily matches the definition of the -th entry of .

Hence the matrices and must be equal!

Recall that in contrast, the proof for the inverses was much easier. The proof for the * operation should now be easy!