What are some specific operations for matrices, like is dot product for vectors?

I read this question and the associated answer (What is the dot product between a vector of matrices?), and it wasn't what I am looking for.

I want to know if there are some special operations designed for bi-dimensional or multi-dimensional vectors ?

When I refer to dot product, I don't mean the equivalent for matrices, but rather an operation that deals with matrices and computes something.

• Both matrix addition and matrix multiplication are defined, both taking two matrices and returning a matrix. The determinant of a matrix is also defined, taking a single matrix and returning a number. – user247327 Mar 31 '16 at 12:53

There are many specific operations for matrices, usually given as a bilinear product from $M_n(K)\times M_n(K)\rightarrow M_n(K)$, like the matrix product or the Lie bracket $(X,Y)\mapsto [X,Y]=XY-YX$. Another operation is the Kronecker product from $M_{m,n}(K)\otimes M_{p,q}(K)\rightarrow M_{mp,nq}(K)$.