are defined only when the blocks have compatible dimensions. One can also multiply supermatrices by elements of R (on the left or right), however, this operation differs from the ungraded case due to the presence of odd elements in R. Let Mr|s×p|q(R) denote the set of all supermatrices over R with dimension (r|s)×(p|q). This set forms a supermodule over R under supermatrix addition and scalar multiplication. In particular, if R is a superalgebra over a field K then Mr|s×p|q(R) forms a super vector space over K. Let Mp|q(R) denote the set of all square supermatices over R with dimension (p|q)×(p|q). This set forms a superring under supermatrix addition and multiplication. Furthermore, if R is a commutative superalgebra, then supermatrix multiplication is a bilinear operation, so that Mp|q(R) forms a superalgebra over R. Addition Two supermatrices of dimension (r|s)×(p|q) can be added just as in the ungraded case to obtain a supermatrix of the same dimension. The addition can be performed blockwise since the blocks have compatible sizes. It is easy to see that the sum of two even supermatrices is even and the sum of two odd supermatrices is odd. Multiplication One can multiply a supermatrix with dimensions (r|s)×(p|q) by a supermatrix with dimensions (p|q)×(k|l) as in the ungraded case to obtain a matrix of dimension (r|s)×(k|l). The multiplication can be performed at the block level in the obvious manner: Note that the blocks of the product supermatrix Z = XY are given by If X and Y are homogeneous with parities |X| and |Y| then XY is homogeneous with parity |X| + |Y|. That is, the product of two even or two odd supermatrices is even while the product of an even and odd supermatrix is odd. Scalar multiplication Scalar multiplication for supermatrices is different than the ungraded case due to the presence of odd elements in R. Let X be a supermatrix. Left scalar multiplication by α ∈ R is defined by where the internal scalar multiplications are the ordinary ungraded ones and denotes the grade involution in R. This is given on homogeneous elements by Right scalar multiplication by α is defined analogously: If α is even then and both of these operations are the same as the ungraded versions. If α and X are homogeneous then α·X and X·α are both homogeneous with parity |α| + |X|. Furthermore, if R is supercommutative then one has As