the (p|q)×(r|s) supermatrix where At denotes the ordinary transpose of A. This can be extended to arbitrary supermatrices by linearity. Unlike the ordinary transpose, the supertranspose is not generally an involution, but rather has order 4. Applying the supertranspose twice to a supermatrix X gives If R is supercommutative, the supertranspose satisfies the identity Parity transpose The parity transpose of a supermatrix is a new operation without an ungraded analog. Let be a (r|s)×(p|q) supermatrix. The parity transpose of X is the (s|r)×(q|p) supermatrix That is, the (i,j) block of the transposed matrix is the (1−i,1−j) block of the original matrix. The parity transpose operation obeys the identities as well as where st denotes the supertranspose operation. Supertrace The supertrace of a square supermatrix is the Z2-graded analog of the trace. It is defined on homogeneous supermatrices by the formula where tr denotes the ordinary trace. If R is supercommutative, the supertrace satisfies the identity for homogeneous supermatrices X and Y. Berezinian The Berezinian (or superdeterminant) of a square supermatrix is the Z2-graded analog of the determinant. The Berezinian is only well-defined on even, invertible supermatrices over a commutative superalgebra R. In this case it is given by the formula where det denotes the ordinary determinant (of square matrices with entries in the commutative algebra R0). The Berezinian satisfies similar properties to the ordinary determinant. In particular, it is multiplicative and invariant under the supertranspose. It is related to the supertrace by the formula References Matrices Super linear algebra