Row Equivalent Matrices
Two matrices are considered row equivalent if one can be transformed into the other through a series of row operations. These operations are defined as follows:
- Add a non-zero scalar multiple \( k \neq 0 \) of row \( R_j \) to row \( R_i \) (where \( i \neq j \)): $$ R_i + R_j \cdot k $$
- Swap the positions of any two rows in the matrix: $$ R_i \Leftrightarrow R_j $$
Applications: Row equivalence is used primarily to convert a matrix into row echelon form, a key step in solving linear systems. It also provides an alternative method for determining both the rank and the determinant of a matrix.
Example in Practice
Consider the matrix:
$$ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 5 \\ 2 & 6 & 8 \end{pmatrix} $$
Let’s perform a row operation:
$$ R_3 + R_1 \cdot (-2) $$
This means adding row \( R_3 \) to row \( R_1 \) multiplied by -2.
Note: Alternatively, this can be seen as subtracting row \( R_1 \) (scaled by 2) from row \( R_3 \): $$ R_3 - R_1 \cdot 2 $$
We obtain:
$$ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 5 \\ 2-2 & 6-4 & 8-6 \end{pmatrix} $$
which simplifies to:
$$ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 5 \\ 0 & 2 & 2 \end{pmatrix} $$
Next, we swap row \( R_2 \) with row \( R_3 \):
$$ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 2 \\ 0 & 0 & 5 \end{pmatrix} $$
This yields a matrix in row echelon form.
Properties of Row Equivalent Matrices
Row equivalent matrices exhibit the following properties:
- Reflexive Property: Each matrix is row equivalent to itself: $$ A = A $$
- Symmetric Property: If matrix \( A \) is row equivalent to matrix \( B \), then \( B \) is row equivalent to \( A \) as well: $$ A = B \Leftrightarrow B = A $$
- Transitive Property: If matrix \( A \) is row equivalent to matrix \( B \) $$ A = B $$, and matrix \( B \) is row equivalent to matrix \( C \) $$ B = C $$, then matrix \( A \) is row equivalent to matrix \( C \): $$ A = C $$
These properties form a foundational framework for further applications in linear algebra.
