Double Entry Bookkeeping and Error Correction

We formally establish the link between linear algebra and the double entry bookkeeping system: the system transforms a vector of numerical values corresponding to transaction amounts to a financial statements vector through matrix multiplication. The matrix is called a generator matrix (it is used to generate financial statements). This representation of double entry bookkeeping allows us to simultaneously apply coding theory and linear algebra's nullspace to the analysis of the relationship between transactions and account balances. We show how to construct a parity check matrix directly from the generator matrix. Our main results demonstrate how the parity check matrix can be used to detect and correct errors which can not be corrected and may not even be detected by the usual trial balance check. (The trial balance check would be a subset of possible checks performed via the parity check matrix.) The idea is that there are more restrictions on the relations among the transaction which are captured by the generator matrix. These restrictions are then exploited by the parity check matrix, allowing for detection and correction of errors.