Computation of the Adjoint Matrix

The best method for computing the adjoint matrix of an order $n$ matrix in an arbitrary commutative ring requires $O(n^{\beta+1/3}\log n \log \log n)$ operations, provided the complexity of the algorithm for multiplying two matrices is $\gamma n^\beta+o(n^\beta)$. For a commutative domain -- and under the same assumptions -- the complexity of the best method is ${6\gamma n^\beta}/{(2^{\beta}-2)}+o(n^\beta)$. In the present work a new method is presented for the computation of the adjoint matrix in a commutative domain. Despite the fact that the number of operations required is now 1.5 times more, than that of the best method, this new method permits a better parallelization of the computational process and may be successfully employed for computations in parallel computational systems.