The BackwardSubstitute function applies the backward substitution described by the upper triangular part of the square mod m Matrix A to the mod m Matrix or Vector B.

Note: It is assumed that A is in upper triangular form, or that only the upper triangular part is relevant, as the lower triangular part of A is ignored.

The mod m Matrix or Vector B must have the same number of rows as there are columns of A.

Application of backward substitution requires that m is a prime, but in some cases it can be computed when m is composite. In cases where it cannot be computed for m composite, a descriptive error message is returned.

The BackwardSubstitute function is most often used in combination with LUDecomposition, and is used in LUApply.

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form BackwardSubstitute(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][BackwardSubstitute](..).