Solving Ax=b by Augmented Row Reduction

From \(A\mathbf{x}=\mathbf{b}\) to an Augmented Matrix

A system written as \(A\mathbf{x}=\mathbf{b}\) records several scalar equations at once. Here \(A\) is an \(m\times n\) real matrix, \(\mathbf{x}\in\mathbb{R}^n\) is the column of unknowns, and \(\mathbf{b}\in\mathbb{R}^m\) is the column of constants. Solving the system means finding values of the unknowns for which every equation holds. Augmented row reduction gives a compact way to carry out the same equation operations on all rows at once.

The coefficient matrix \(A\) contains the coefficients of the unknowns; the extra column \(\mathbf{b}\) contains the right-hand sides. The vertical bar is a visual separator, not an additional column of coefficients. Since each row represents an equation, any row operation used to simplify the system must be applied to the entire row, including the entry to the right of the bar.

These operations correspond to familiar equation operations: reorder equations, multiply an equation by a nonzero number, or add a multiple of one equation to another. In every case, the right-hand sides must undergo the same operation as the coefficients. The goal is often to reach row-echelon form: all zero rows, if any, are at the bottom, and the first nonzero entry in each nonzero row lies farther to the right than the first nonzero entry in the row above it. Those first nonzero entries are called leading entries. When the leading entries identify the variables in order, the equations can be solved from the bottom row upward by back-substitution.

Why Row Operations Preserve Solutions

Proof. A row of the augmented matrix represents an equation. Swapping two rows changes only the order of the equations, so a vector \(\mathbf{x}\) satisfies all the equations before the swap exactly when it satisfies all of them after the swap. Multiplying a row by a nonzero scalar \(c\) replaces an equation \(L(\mathbf{x})=d\) by \(cL(\mathbf{x})=cd\). If the original equation holds, the new one holds; if the new equation holds, division by \(c\ne0\) recovers the original. Finally, replacing one row by itself plus \(c\) times another replaces \(L_1(\mathbf{x})=d_1\) by \(L_1(\mathbf{x})+cL_2(\mathbf{x})=d_1+cd_2\), while leaving the second equation in the system. If both original equations hold, the replacement equation holds. Conversely, subtracting \(c\) times the unchanged second equation recovers the original first equation. Thus each operation preserves the solutions in both directions. \(\square\)

The conclusion also applies to a sequence of row operations.

Proof. For a sequence of zero operations, the matrix is unchanged, so its system has the same solutions. For a sequence of \(k\ge1\) operations, the first \(k-1\) operations preserve the solution set by induction on the sequence length. The final operation preserves it by the theorem. Therefore the full sequence preserves the solution set. \(\square\)

This is the justification for row-reducing an augmented matrix: the simplified equations are not an approximation or a different problem. They are equivalent equations, obtained by reversible steps. Once the system has a form that can be solved directly, the values found must satisfy the original system as well. Substitution is a useful independent check on the arithmetic.

A Reliable Row-Reduction Routine

Row reduction does not require every leading entry to be \(1\); scaling a row can make back-substitution easier, but it is optional. The essential discipline is to record the operation correctly and apply it to every entry in the row. The examples below use small systems so that each intermediate entry can be checked directly.

Worked Systems

Common Pitfall: Losing Track of the Right-Hand Side

A frequent error is to update the coefficients in a row but calculate its right-hand side incorrectly. For instance, begin with rows \([2,1\mid5]\) and \([1,-1\mid1]\), and apply \(R_2\to R_2-2R_1\). The coefficient entries become \(1-2(2)=-3\) and \(-1-2(1)=-3\), while the right-hand side becomes \(1-2(5)=-9\). Therefore the correct new row is \([-3,-3\mid-9]\), not \([0,-3\mid-9]\).

The first coefficient does become \(0\) if the rows are swapped first. After the swap, the rows are \([1,-1\mid1]\) and \([2,1\mid5]\). Applying \(R_2\to R_2-2R_1\) now gives \([0,3\mid3]\): the entries are \(2-2(1)=0\), \(1-2(-1)=3\), and \(5-2(1)=3\). The operation and the row order must match the displayed calculation.

A second version of the same pitfall is changing a coefficient row but copying the old right-hand side. This does not represent an equivalent equation. The theorem above explains why: the right-hand side is part of the equation and must undergo the same row operation. When in doubt, calculate each new entry, including the final column, from the stated operation.

What to Keep in Mind

Augmented row reduction is a bookkeeping method grounded in equivalence of equations. Set up the coefficient columns in a fixed variable order, include the entire right-hand-side column, and write each operation so that its effect can be checked. After reduction, use the resulting equations to determine the values, then substitute those values into the original system. The substitution check tests the final answer; checking the intermediate rows tests whether the reduction itself was carried out correctly.

In the worked systems, the reduced rows directly specified the values of all variables. Other row-echelon forms require additional care in interpreting the equations; the first task is always to read each row as an equation in the same ordered variables. The preservation theorem remains the reason the reduction is legitimate in every case.