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Matrices and systems of linear equations

Row-echelon form

A matrix is in row-echelon form if:

The leading (leftmost non-zero) entry of a row (if any) is called a pivot.

A matrix is in reduced row-echelon form if it is in row-echelon form and


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Elementary row operations

The following three operations are called elementary row operations:

Matrix A is row-equivalent to matrix B if B is obtained from A by a sequence of elementary row operations.


Given the matrix img; R2R2 + 2R3 img; R2R3 img; R12R1 imgA and B are row-equivalent.

It's not difficult to prove that if A and B are row-equivalent augmented matrices of two systems of linear equations, then the two systems have the same solutions set: solving one of the two systems is exactly the same thing.

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The Gauss-Jordan algorithm

This is a process that starts with a given matrix A and produces a matrix B in (reduced) row-echelon form, which is row-equivalent to A. Reduced row-echelon form is better, but row-echelon form is enough for almost all purposes. If A is the augmented matrix of a system of linear equations, then B will be a much simpler matrix than A from which the consistency or inconsistency of the corresponding system is immediately apparent and in fact the complete solution of the system can be read off. As we want to apply this process to the solution of linear systems we assume that there are no zero rows in the original augmented matrix (a zero row means a meaningless equation).

This process is made up of the following steps:

  1. Step 1.  Find the first non-zero column moving from left to right (usually there are no zero columns, otherwise the corresponding unknown is useless, but it is better to consider this more general case).
  2. Step 2. By interchanging rows, if necessary, ensure that the first entry in this column is non-zero: this leading entry is the first pivot, p1.
  3. Step 3. Use the third elementary row operation to obtain all zeros "under" this pivot. You can proceed as follows: if the row number i has a non zero entry, say a, under the pivot, multiply the first row by img and add it to row number i.
  4. Step 4. Repeat the process to the matrix you obtain by deleting row 1, until you obtain the row-echelon form.
  5. Step 5. If you are interested in the reduced row-echelon form, divide each row by the pivot and suitably use the third elementary row operation.
See an example.

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Systematic solution of linear systems

Suppose a system of m linear equations in n unknowns x1, x2, ..., xn has augmented matrix A|b. Via the Gauss-Jordan algorithm transform the matrix A|b in a matrix B|c with B in row-echelon (or reduced row-echelon) form. The number of pivots in the matrix B is called the rank, r, of the system, and we must have rmin(m,n).

Once the augmented matrix is so reduced the consistency or inconsistency of the system can immediately be checked:

Also the number of solutions can immediately be checked.

See the examples.

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Homogeneous systems

If all constants are zero the system is called homogeneous. A homogenous system is always consistent as it has at least the trivial solution (0, 0, ... ,0). It also has other non trivial solutions if m<n.

Homogeneous systems play an important role in the theory of linear functions between vector spaces.

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first published on march 15 2002 - last updated on september 01 2003