In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinary field (thought of as a purely even commutative superalgebra). Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry. Definitions and notation Let R be a fixed superalgebra (assumed to be unital and associative). Often one requires R be supercommutative as well (for essentially the same reasons as in the ungraded case). Let p, q, r, and s be nonnegative integers. A supermatrix of dimension (r|s)×(p|q) is a matrix with entries in R that is partitioned into a 2×2 block structure with r+s total rows and p+q total columns (so that the submatrix X00 has dimensions r×p and X11 has dimensions s×q). An ordinary (ungraded) matrix can be thought of as a supermatrix for which q and s are both zero. A square supermatrix is one for which (r|s) = (p|q). This means that not only is the unpartitioned matrix X square, but the diagonal blocks X00 and X11 are as well. An even supermatrix is one for which the diagonal blocks (X00 and X11) consist solely of even elements of R (i.e. homogeneous elements of parity 0) and the off-diagonal blocks (X01 and X10) consist solely of odd elements of R. An odd supermatrix is one for which the reverse holds: the diagonal blocks are odd and the off-diagonal blocks are even. If the scalars R are purely even there are no nonzero odd elements, so the even supermatices are the block diagonal ones and the odd supermatrices are the off-diagonal ones. A supermatrix is homogeneous if it is either even or odd. The parity, |X|, of a nonzero homogeneous supermatrix X is 0 or 1 according to whether it is even or odd. Every supermatrix can be written uniquely as the sum of an even supermatrix and an odd one. Algebraic structure Supermatrices of compatible dimensions can be added or multiplied just as for ordinary matrices. These operations are exactly the same as the ordinary ones with the restriction that they