Row Operations of Matrices

Understand elementary row operations and how they affect matrices and determinants

What are Elementary Row Operations?

Elementary row operations are the basic transformations you can apply to the rows of a matrix. They are essential for Gaussian elimination, Gauss-Jordan inversion, and solving systems of linear equations.

Definition:

An elementary row operation is an operation that transforms a matrix by changing its rows while preserving the equivalence of the represented linear system.

The Three Elementary Row Operations

  • Row replacement: Add a multiple of one row to another
  • Row scaling: Multiply a row by a non-zero scalar
  • Row interchange: Swap two rows
Why It Matters:

These operations can be performed manually, but they can also be done by matrix multiplication using specially constructed elementary matrices.

Elementary Matrices

Each elementary row operation can be represented as multiplication by an elementary matrix. An elementary matrix is obtained by applying a single row operation to an identity matrix.

General idea:
Apply row operation to identity matrix \(I\) to get an elementary matrix \(E\).
Then: \(EA\) applies that row operation to matrix \(A\).
Note:

Left-multiplying by an elementary matrix performs a row operation. Right-multiplying would correspond to a column operation.

Operation 1: Add a Multiple of One Row to Another

Row Replacement

Type I

Add \(k\) times row 2 to row 3. This is done by placing \(k\) in row 3, column 2 of the identity matrix.

  • Start with identity matrix \(I\)
  • Place \(k\) in position (3,2)
  • Multiply \(E \cdot A\) to apply the row replacement
\(\displaystyle \left[\begin{matrix}1&0&0\\0&1&0\\0&k&1\end{matrix}\right] \cdot \left[\begin{matrix}a&b&c\\d&e&f\\g&h&i\end{matrix}\right] = \left[\begin{matrix}a&b&c\\d&e&f\\kd+g&ke+h&kf+i\end{matrix}\right]\)
Determinant Effect:

This row operation does not change the determinant.

Operation 2: Multiply a Row by a Non-Zero Scalar

Row Scaling

Type II

Multiply row 2 by a non-zero scalar \(k\). This is represented by placing \(k\) on the diagonal of the identity matrix at position (2,2).

\(\displaystyle \left[\begin{matrix}1&0&0\\0&k&0\\0&0&1\end{matrix}\right] \cdot \left[\begin{matrix}a&b&c\\d&e&f\\g&h&i\end{matrix}\right] = \left[\begin{matrix}a&b&c\\kd&ke&kf\\g&h&i\end{matrix}\right]\)
Determinant Effect:

This row operation multiplies the determinant by \(k\).

Operation 3: Interchange Two Rows

Row Interchange

Type III

Swap row 2 and row 3. This is represented by swapping those two rows of the identity matrix.

\(\displaystyle \left[\begin{matrix}1&0&0\\0&0&1\\0&1&0\end{matrix}\right] \cdot \left[\begin{matrix}a&b&c\\d&e&f\\g&h&i\end{matrix}\right] = \left[\begin{matrix}a&b&c\\g&h&i\\d&e&f\end{matrix}\right]\)
Determinant Effect:

Swapping two rows changes the sign of the determinant: \(\det(A') = -\det(A)\)

Summary: Effects on Determinants

Operation Effect on Determinant
Row replacement (add multiple) No change
Row scaling (multiply by \(k\)) Multiplies determinant by \(k\)
Row interchange (swap rows) Changes sign

Key Points to Remember

  • Row operations are fundamental for solving systems and finding inverses
  • Each elementary row operation can be represented by an elementary matrix \(E\)
  • Left multiplication \(EA\) applies row operations to \(A\)
  • Row replacement does not change the determinant
  • Row scaling multiplies the determinant by the scalar
  • Row interchange flips the sign of the determinant
  • These properties are used heavily in Gaussian elimination
















Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?