Why do large linear systems accumulate rounding errors?

In numerical analysis, large linear systems accumulate rounding errors primarily due to the finite length of numbers used in standard computer arithmetic.1

When solving systems of linear equations—typically represented as $Ax=b$—computers must represent numbers with a limited number of digits. Because these systems often require a large number of computational steps to reach a solution, these small, inherent inaccuracies in numerical representation compound over the course of the calculation. While direct methods like Gaussian elimination are theoretically exact in a finite number of steps, the practical implementation on digital hardware inevitably introduces these rounding errors, leading to discrepancies between the computed solution and the true mathematical result.12