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Elementary
Matrices

Discover the building blocks of row reduction: how a single row operation becomes a matrix you can multiply, invert, and compose.

Linear Algebra

What Is an Elementary Matrix?

An elementary matrix is a matrix obtained by performing exactly one elementary row operation on the identity matrix \( I \).

This idea is powerful: it turns every step of Gaussian elimination into a matrix multiplication.

The Three Row Operations

Type I: Swap
Interchange two rows: \( R_i \leftrightarrow R_j \)
Type II: Scale
Multiply a row by a nonzero scalar: \( R_i \to cR_i \), \( c \neq 0 \)
Type III: Add
Add a scalar multiple of one row to another: \( R_i \to R_i + cR_j \)

From \( I \) to \( E \)

Identity \( I_3 \)
\( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
\( R_1 \leftrightarrow R_2 \)
Type I
\( \begin{pmatrix} \color{#818cf8}{0} & \color{#818cf8}{1} & \color{#818cf8}{0} \\ \color{#818cf8}{1} & \color{#818cf8}{0} & \color{#818cf8}{0} \\ 0 & 0 & 1 \end{pmatrix} \)
\( R_2 \to 5R_2 \)
\( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
Type II
\( \begin{pmatrix} 1 & 0 & 0 \\ 0 & \color{#4ade80}{5} & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
Type III
\( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
\( R_3 \to R_3 + 4R_1 \)
\( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \color{#fbbf24}{4} & 0 & 1 \end{pmatrix} \)

Each matrix above was built by doing one row operation to \( I_3 \). The changed entries are highlighted.

Core Theorem

This means Gaussian elimination on \( A \) can be written as a product of elementary matrices times \( A \): $$ E_k \cdots E_2 E_1 A = R $$ where \( R \) is the row-echelon form.

Every \( E \) Is Invertible

Each row operation can be undone by another row operation of the same type. Therefore every elementary matrix has an inverse, and that inverse is also elementary.

1
Swap \( R_i \leftrightarrow R_j \): undo by swapping again. \( E^{-1} = E \).
2
Scale \( R_i \to cR_i \): undo by scaling by \( \frac{1}{c} \). So \( E^{-1} \) replaces \( c \) with \( \frac{1}{c} \).
3
Add \( R_i \to R_i + cR_j \): undo by subtracting. \( E^{-1} \) replaces \( c \) with \( -c \).

Operation vs Matrix

Row Operation Elementary Matrix
What it is A verb: an action on rows A noun: a specific matrix
How it acts Directly manipulates entries Left-multiplies: \( EA \)
Reversibility Undo with opposite operation Multiply by \( E^{-1} \)
Composition Do operations in sequence Multiply matrices: \( E_2 E_1 \)

How to Build \( E \)

1
Identify the row operation you want to perform on matrix \( A \).
2
Start with the identity matrix \( I_n \) of the same number of rows as \( A \).
3
Apply that single row operation to \( I_n \). The result is \( E \).
4
Verify by checking that \( EA \) gives the same result as performing the operation directly on \( A \).

Example 1: Row Swap

Perform \( R_1 \leftrightarrow R_2 \) on \( A = \begin{bmatrix} 3 & 1 \\ 7 & 2 \end{bmatrix} \).

Example 2: Row Scaling

Perform \( R_2 \to 3R_2 \) on \( A = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 6 & 7 \end{bmatrix} \).

Example 3: Row Addition

Perform \( R_3 \to R_3 + 4R_1 \) on \( A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 5 \end{bmatrix} \).

Factoring \( A \) as a Product

If \( A \) is invertible, row reduction takes \( A \) to \( I \):

This is one of the most important results in linear algebra: invertibility, row reduction, and elementary matrices are all deeply connected.

Determinants of \( E \)

Type I: Swap
\( \det(E) = -1 \)
Type II: Scale by \( c \)
\( \det(E) = c \)
Type III: Add
\( \det(E) = 1 \)

Composing Operations

A
\( E_1 A \)
\( E_1 A \)
\( E_2(\cdots) \)
\( E_2 E_1 A \)
\( \cdots \)
 R Row Echelon Form

Order matters: The most recent operation is the leftmost matrix. Read right-to-left: \( E_1 \) acts first, then \( E_2 \), and so on.

Common Mistakes

✕ Wrong Multiplication Side
Elementary matrices left-multiply. Writing \( AE \) instead of \( EA \) gives a column operation, not a row operation.
✕ Wrong Size of \( I \)
Use the identity matching the number of rows of \( A \), not columns. For an \( m \times n \) matrix, \( E \) is \( m \times m \).
✕ Confusing Inverse Signs
For Type III, the inverse negates the scalar (\( c \to -c \)). For Type II, it reciprocates (\( c \to 1/c \)). Do not mix these up.

Summary Table

Property Type I (Swap) Type II (Scale) Type III (Add)
Operation \( R_i \leftrightarrow R_j \) \( R_i \to cR_i \) \( R_i \to R_i + cR_j \)
Form of \( E \) Swap rows \( i,j \) of \( I \) Put \( c \) at \( (i,i) \) Put \( c \) at \( (i,j) \)
Inverse \( E^{-1} = E \) Replace \( c \) with \( \frac{1}{c} \) Replace \( c \) with \( -c \)
\( \det(E) \) \( -1 \) \( c \) \( 1 \)

Question 1 of 5

Which matrix is the elementary matrix for the row operation \( R_1 \leftrightarrow R_3 \) on a \( 3 \times 3 \) matrix?

Question 2 of 5

What is the inverse of the elementary matrix \( E = \begin{bmatrix} 1 & 0 \\ 0 & 5 \end{bmatrix} \)?

Question 3 of 5

What is \( \det(E) \) if \( E \) performs the operation \( R_2 \to R_2 + 7R_1 \)?

Question 4 of 5

If \( E_3 E_2 E_1 A = I \), which of the following equals \( A \)?

Question 5 of 5

Let \( E = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -3 & 0 & 1 \end{bmatrix} \). What row operation does left-multiplying by \( E \) perform?