  We give the criterion for the irreducibility, the Schur irreducibility and
the indecomposability of the set of two $n\times n$ matrices $\Lambda_n$ and
$A_n$ in terms of the subalgebra associated with the "support" of the matrix
$A_n$, where $\Lambda_n$ is a diagonal matrix with different non zeros
eigenvalues and $A_n$ is an arbitrary one. The list of all maximal subalgebras
of the algebra ${\rm Mat}(n,{\mathbb C})$ and the list of the corresponding
invariant subspaces connected with these two matrices is also given. The
properties of the corresponding subalgebras are expressed in terms of the
graphs associated with the support of the second matrix.
  For arbitrary $n$ we describe all minimal subsets of the elementary matrices
$E_{km}$ that generate the algebra ${\rm Mat}(n,{\mathbb C})$.
