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The Gauss-Jordan algorithm.

We can perform three types of elementary operations on the rows of a matrix:

1.
Interchange two rows: \fbox{$R_i \leftrightarrow R_j$ .}
2.
Multiply a row by a non-zero scalar: \fbox{$R_i \leftarrow \alpha R_i$ .}
3.
Add a row to another row: \fbox{$R_i \leftarrow R_i+R_j$ }.
A combination of the two last operations is as follows: Add to a row a multiple of another row: \fbox{$R_i \leftarrow R_i+\alpha R_j$ }.

Definition 3.2.1   A matrix B is row equivalent to a matrix A if B can be obtained from A by successive elementary row operations.

Proposition 3.2.2   If a matrix B can be obtained from a matrix A by successive elementary row operations, then the reverse process is possible too, i.e. A can be obtained from B by successive elementary row operations.

Example 3.2.3   On the one-hand, we have:

\begin{displaymath}\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatr...
...gin{pmatrix}1 & 2 & 3 \\ 0 & -3 & -6 \\ 7 & 8 & 9 \end{pmatrix}\end{displaymath}

On the other hand, we have:

\begin{displaymath}\begin{pmatrix}1 & 2 & 3 \\ 0 & -3 & -6 \\ 7 & 8 & 9 \end{pma...
...begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\end{displaymath}

What we showed here for a single elementary operation is true for any finite sequence of elementary operations.

Proposition 3.2.4   Two row equivalent matrices are the augmented matrices of two systems of linear equations having the same set of solutions.

In such a case, we say that the systems are equivalent.

The Gauss-Jordan algorithm:

\fbox{
\begin{minipage}{12cm}
\begin{enumerate}
\item If they are zero-rows, pus...
...ottom towards top'' and
\lq\lq from right to left''.
\end{enumerate}\end{minipage} }

Example 3.2.5  
\begin{align*}\begin{pmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{pmatrix} \overrightarrow{...
...tarrow R_1 - 2R_2}\begin{pmatrix}1&0&-1\\ 0&1&2\\ 0&0&0\end{pmatrix}\end{align*}
The last matrix is row-reduced.

Example 3.2.6  
\begin{align*}\begin{pmatrix}
1 & 2 & -1 & 1 & 1 \\ 1 & 2 & 3 & 0 & 1 \\ 1 & 2 &...
...ldsymbol{1} & 0 & 1 \\
0 & 0 & 0 & \boldsymbol{1} & 1
\end{pmatrix}\end{align*}
The last matrix is row-reduced.

Proposition 3.2.7   For any matrix A, there exists a unique row-reduced matrix B row-equivalent to A.

This means that, even if you row-reduce a matrix by two different ways (by means of elementary row operations), the row-reduced matrix you get is the same.


next up previous contents
Next: Solution of a system Up: Systems of linear equations. Previous: Systems of equations and
Noah Dana-Picard
2001-02-26