When two block matrices have the same shape and their diagonal blocks are square matrices, then they multiply similarly to matrix
multiplication. For example,

(7)

Note that the usual rules of matrix multiplication hold even when the block matrices are not square (assuming that the block sizes correspond). Of course, matrix multiplication is in general not commutative, so in these block matrix multiplications, it is important to keep the correct order of the multiplications.

When the blocks are square matrices, the set of invertible block matrices is a
group isomorphic to the general linear group , where is the ring of square matrices.