Maintaining Exact Arithmetic While Solving Ax=b

Why Exact Arithmetic Matters

When a system has integer or fractional coefficients, row reduction can create fractions even if none appeared in the original equations. Rounding those fractions may seem harmless, but a rounded entry can turn a nonzero pivot into zero, or conceal a contradiction in the last row. Either error changes the conclusion about the system: it may appear to have fewer pivots than it really does, or to be consistent when it is not.

For this tutorial, assume the entries of the coefficient matrix \(\mathbf{A}\) and right-hand side \(\mathbf{b}\) are rational numbers. Write each rational number as a fraction and retain it exactly during the calculation. In particular, do not replace a fraction by a rounded decimal partway through row reduction. A finite decimal can represent an exact rational number if it is specified as exact; the danger is rounding an exact value and then treating the rounded value as though it were unchanged.

The three elementary row operations are swapping two rows, multiplying a row by a nonzero scalar, and adding a scalar multiple of one row to another. The goal is not to avoid fractions; it is to calculate them correctly. A pivot is an entry used to eliminate entries in its column. With exact arithmetic, a pivot is zero precisely when its numerator is zero after the fraction is simplified. A small nonzero fraction is still a valid pivot.

Why Exact Row Operations Preserve the Answers

Proof. Each row of an augmented matrix represents an equation in the unknown entries of \(\mathbf{x}\). Swapping two rows only changes the order of two equations. Multiplying a row by a nonzero scalar replaces an equation by an equivalent equation, because we can divide by that same scalar to recover the original. Adding a multiple of one row to another also preserves equivalence: subtracting that multiple reverses the operation. Therefore each elementary row operation preserves the set of vectors satisfying all the equations. A finite sequence of such operations preserves the same set at every step. Finally, exact addition, subtraction, multiplication, and division by a nonzero rational number produce rational numbers, so carrying out these operations without rounding retains the actual entries at each step. \(\square\)

This theorem explains why exactness matters to the method. Row reduction makes equivalent equations; rounding an entry is not one of the elementary row operations. It can replace the current system with a different one. In the calculations below, each row-operation chain therefore uses fractions rather than rounded approximations.

Exact Pivots and Consistency

After row reduction, a row with zeros in all coefficient columns but a nonzero right-hand side represents an equation of the form \(0=c\), where \(c\ne0\). Such an equation cannot be satisfied, so the system is inconsistent. Conversely, if a row-reduced augmented matrix contains no such row, the reduced equations do not include a contradiction; solving them by assigning values to any free variables gives solutions. This is why exact zero tests in the augmented matrix matter.

Proof. If such a row occurs, its equation is \(0=c\) with \(c\ne0\), which no vector of unknowns can satisfy. Thus the system is inconsistent. For the converse, suppose no such row occurs. Every nonzero row then has a nonzero coefficient somewhere, and every all-zero coefficient row has right-hand side zero. Assign arbitrary values to the variables in columns without pivots. Starting with the last pivot row and proceeding upward, solve each pivot equation for its pivot variable: its coefficient is nonzero, and the variables to its right already have assigned values. Rows with all coefficients zero impose only \(0=0\). These assignments satisfy every row, so the reduced system has a solution. Since row operations preserve the solution set, the original system is consistent. \(\square\)

A Reliable Exact-Arithmetic Routine

When the entries are rational, a careful workflow keeps each operation auditable and prevents rounding from influencing the answer.

There are two different issues that should not be confused. If the problem gives exact rational data, exact fraction arithmetic determines the answer for that system. If the numbers came from measurements and were only recorded approximately, treating the recorded decimals as exact does not recover the unknown true data; it only solves the system represented by those recorded values. The exact method prevents arithmetic rounding from adding a further, avoidable change.

Common Mistakes

Calling a small fraction zero. In the second worked example, replacing \(\tfrac{1}{1000}\) by \(0\) makes a false free variable and loses the unique solution. The correction is to keep the fraction exact and use it as a pivot, or multiply its row by \(1000\) to obtain a pivot of \(1\).

Rounding each row operation. A fraction created in one step may affect a later pivot or right-hand side. Rounding it immediately can change both. Retain the fraction through the whole reduction; simplify by common factors only when the equality is exact.

Assuming every system has a solution. A row of the form \(\begin{bmatrix}0&0&|&1\end{bmatrix}\) is not a request to choose a variable value. It states the impossible equation \(0=1\). Preserve the right-hand-side entry exactly so that a genuine contradiction is not rounded away.

Exact arithmetic is not a separate method for solving systems. It is a way to carry out the familiar augmented row reduction without silently replacing the system by a nearby one. The distinction is most important when a pivot or a contradiction depends on a small rational value.