No, Trial Balance is not a complete proof of arithmetical accuracy of account. A Trial Balance in which the credit and debit accounts match does not prove that, all transactions have been recorded in the proper accounts. For example, the wages paid for the installation of machinery had been erroneously recorded by debiting the wages account in the place of machinery account, the Trial Balance would still agree.
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Similarly, an agreed Trial Balance does not prove that all transactions have been recorded in the books of original entry. For example, a credit sale invoice were to be completely omitted from being recorded in the sales day book, the error would not be disclosed in the Trial Balance.
To conclude, we can say that a trial balance should not be recorded as a conclusive proof of the correctness of the books of account.