Basics of elementary row operations行基本変形ぎょうきほんへんけい

date2026-07-14document_iddoc_8ae033f0135b9133968f42e9d959282cdescription行基本変形を、連立一次方程式の解集合を保ったまま係数を整理する三種類の操作として説明し、左から基本行列を掛ける意味と列基本変形との違いまで整理する講義である。prerequisites拡大係数行列 / 行列の基本演算type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

An elementary row operation行基本変形ぎょうきほんへんけい is a central operation for solving a system of linear equations連立一次方程式れんりついちじほうていしき. Its purpose is to make the coefficients easier to read without changing the solution set.

Why do we consider these operations? Because each row operation replaces equations by equivalent equations. Later they are used in elimination, rank, inverse matrices, and determinant computation, but the basic reason is that they preserve the solution set.

2Terms and definitions

Elementary row operations are the following three reversible operations on rows.

\begin{array}{c|c|c} \text{Operation} & \text{Notation} & \text{Meaning}\\ \hline \text{Row swap} & R_i\leftrightarrow R_j & \text{exchange row }i\text{ and row }j\\ \text{Nonzero row scaling} & R_i\leftarrow cR_i,\ c\ne 0 & \text{multiply row }i\text{ by }c\\ \text{Row replacement} & R_i\leftarrow R_i+cR_j & \text{add }c\text{ times row }j\text{ to row }i \end{array}\begin{array}{c|c|c} \text{Operation} & \text{Notation} & \text{Meaning}\\ \hline \text{Row swap} & R_i\leftrightarrow R_j & \text{exchange row }i\text{ and row }j\\ \text{Nonzero row scaling} & R_i\leftarrow cR_i,\ c\ne 0 & \text{multiply row }i\text{ by }c\\ \text{Row replacement} & R_i\leftarrow R_i+cR_j & \text{add }c\text{ times row }j\text{ to row }i \end{array}

The condition c != 0 is necessary because undoing the scaling uses 1/c. Since we divide by a symbolic quantity here, the nonzero condition must be stated explicitly.

3Plan

There are two viewpoints on row operations.

The first is the equation viewpoint. Each row represents one equation. Swapping equations, multiplying an equation by a nonzero scalar, or adding a multiple of one equation to another creates an equivalent system with the same solutions.

The second is the matrix viewpoint. An elementary row operation is multiplication by an elementary matrix基本行列きほんぎょうれつ from the left.

A ⟼ E A .

This formula means that the matrix on the left forms new rows as linear combinations of the rows of $A$ .

4Intuitive explanation

In a system of linear equations, changing the order of equations does not change the solutions. Multiplying an entire equation by a nonzero scalar only writes the same condition using a different scale. Adding a multiple of an already required equation to another equation also keeps the same solution set.

Thus elementary row operations clean up the appearance while preserving the solutions. Row reduction produces echelon forms not to change the solutions, but to make the hidden constraints easier to read.

5Worked example

5.1Problem

Use elementary row operations to simplify the augmented matrix and find the corresponding solution.

\left(\begin{array}{cc|c} 1&2&5\\ 3&4&11 \end{array}\right)\left(\begin{array}{cc|c} 1&2&5\\ 3&4&11 \end{array}\right)

5.2Explanation

First apply $R_{2} < - R_{2} - 3 R_{1}$ .

\left(\begin{array}{cc|c} 1&2&5\\ 0&-2&-4 \end{array}\right)\left(\begin{array}{cc|c} 1&2&5\\ 0&-2&-4 \end{array}\right)

Next apply $R_{2} < - - \frac{1}{2} R_{2}$ . The division is by the number $- 2$ , which is nonzero.

\left(\begin{array}{cc|c} 1&2&5\\ 0&1&2 \end{array}\right)\left(\begin{array}{cc|c} 1&2&5\\ 0&1&2 \end{array}\right)

Finally apply $R_{1} < - R_{1} - 2 R_{2}$ .

\left(\begin{array}{cc|c} 1&0&1\\ 0&1&2 \end{array}\right)\left(\begin{array}{cc|c} 1&0&1\\ 0&1&2 \end{array}\right)

Therefore the solution is $x = 1$ , $y = 2$ . This example uses the fact that each row operation preserves the solution set while moving the system to a more readable equivalent form.

6What each operation preserves and changes

As an order note, the main point of this page is that applying row operations to an augmented matrix preserves the solution set. Statements about rank階数かいすう and the determinant行列式ぎょうれつしき are previews of later lectures where these quantities are defined and proved carefully. Here they are named only to show what can change or remain unchanged.

When applied to an augmented matrix, elementary row operations preserve the solution set of the system. For the coefficient matrix alone, they also preserve rank. This is because a row operation is multiplication by an invertible matrix from the left.

However, the actual entries, the particular basis of the row space, and the value of the determinant may change. For a square matrix, the determinant changes as follows.

\begin{array}{c|c} \text{Row operation} & \text{Effect on }\det A\\ \hline R_i\leftrightarrow R_j & \text{changes sign}\\ R_i\leftarrow cR_i & \text{is multiplied by }c\\ R_i\leftarrow R_i+cR_j & \text{does not change} \end{array}\begin{array}{c|c} \text{Row operation} & \text{Effect on }\det A\\ \hline R_i\leftrightarrow R_j & \text{changes sign}\\ R_i\leftarrow cR_i & \text{is multiplied by }c\\ R_i\leftarrow R_i+cR_j & \text{does not change} \end{array}

Thus row operations do not preserve everything. One must distinguish what is preserved and what changes according to the purpose of the calculation.

7Comparison with elementary column operations

An elementary column operation列基本変形れつきほんへんけい acts on columns, so it is represented by multiplication by an elementary matrix from the right.

A ⟼ A F .

Column operations preserve the column space and rank, but they change the meaning of the unknowns in a system $A x = b$ . Therefore row operations are the natural tool when solving systems. If column operations are used, the corresponding change of variables must also be tracked.

8Theorem and proof: row operations preserve the solution set

8.1Theorem

A system obtained from an augmented matrix by elementary row operations has the same solution set as the original system.

8.2Proof

A row swap only changes the order of the equations, so it does not change the solution set.

For nonzero row scaling, the equation $p (x) = q$ is replaced by $c p (x) = c q$ . Since c != 0, multiplying back by 1/c recovers the original equation. Hence the two equations are equivalent.

For row replacement, the equation $p_{i} (x) = q_{i}$ is replaced by $p_{i} (x) + c p_{j} (x) = q_{i} + c q_{j}$ . If $x$ satisfies all original equations, it satisfies the new equation. Conversely, if the new equation and $p_{j} (x) = q_{j}$ both hold, subtracting $c$ times the latter recovers $p_{i} (x) = q_{i}$ . Hence the solution set is unchanged.

9Final form

Elementary row operations are the following three reversible row operations.

R_{i} \leftrightarrow R_{j}, R_{i} \leftarrow c R_{i} (c \neq 0), R_{i} \leftarrow R_{i} + c R_{j} .

For systems of linear equations they preserve the solution set. Statements about rank and determinants are previews of later lectures; at this point, read them as a warning that row operations preserve the target quantity for solving systems but do not preserve every matrix quantity.

10Exercise links

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