AcademyLinear Systems
Academy
Elementary Row Operations
Level 1 - Linear Algebra topic page in Linear Systems.
Principle
Elementary row operations transform a linear system into an equivalent system with the same solution set. They are the legal moves used in Gaussian and Gauss-Jordan elimination.
Each operation changes equations, not the unknown vector that solves them.
Notation
\(R_i\)
row i of a matrix
\(R_i\leftrightarrow R_j\)
swap rows i and j
\(R_i\leftarrow\lambda R_i\)
multiply row i by non-zero scalar \lambda
\(R_i\leftarrow R_i+\lambda R_j\)
add \lambda times row j to row i
\(\lambda\)
a real scalar
\(\sim\)
row-equivalent matrices
The Core Method
There are exactly three elementary row operation types:
- Swap two rows.
- Multiply one row by a non-zero scalar.
- Add a scalar multiple of one row to another row.
These operations preserve the solution set of the corresponding linear system:
Row equivalence
\[[A\mid\mathbf b]\sim[C\mid\mathbf d]\]
The scalar in \(R_i\leftarrow\lambda R_i\) must be non-zero. Multiplying a row by zero destroys information and is not reversible.
Worked Cases
Question
Apply \(R_2\leftarrow R_2-3R_1\) to \(\begin{pmatrix}1&2&|&5\\3&7&|&16\end{pmatrix}\).
Answer
Keep row \(1\) unchanged: \(R_1=(1,2,|,5)\). Compute the new row \(2\): \(R_2-3R_1=(3,7,|,16)-3(1,2,|,5)\). This is \((3-3,7-6,|,16-15)=(0,1,|,1)\). The new matrix is \(\begin{pmatrix}1&2&|&5\\0&1&|&1\end{pmatrix}\).
Examples
Question
Apply \(R_1\leftrightarrow R_3\) to \(\begin{pmatrix}1&0\\2&1\\5&4\end{pmatrix}\).
Answer
Swap the first and third rows while keeping the second row fixed. The result is \(\begin{pmatrix}5&4\\2&1\\1&0\end{pmatrix}\).