Row reduced echelon form of a matrix

Section 1.3 Row reduced echelon form of a matrix

In this section we denote by \(\mathbb{K}\) either the set of real numbers, \(\mathbb{R}\) or the set of complex numbers, \(\mathbb{C}\text{.}\)

Definition 1.3.1. (Elementary row operations).

Let \(A\in M_{m\times n}(\mathbb{K})\text{.}\) Denote by \(R_i\) (for \(1\leq i\leq m\)) the \(i\)-th row of \(A\text{.}\) Following are elementary row operations.

Interchanging \(i\)-th row with \(j\)-th row. This operation is usually denoted by \(R_i\leftrightarrow R_j\text{.}\)
Multiplying \(i\)-th row by a nonzero \(\alpha\in\mathbb{K}\text{.}\) This operation is usually denoted by \(R_i\mapsto \alpha R_i\text{.}\)
Adding a constant multiple of \(j\)-th row to \(i\)-th row. This operation is usually denoted by \(R_i\mapsto R_i+\alpha R_j\text{.}\)

Definition 1.3.2.

A square matrix \(E\in M_n(\K)\) is said to be an elementary matrix if it is obtained from the identity matrix \(I_n\) by performing only one of the row operations described in Definition 1.3.1 on \(I_n\text{.}\)

Note 1.3.3. Procedure to obtain a row reduced echelon form of a matrix.

Recall that given a matrix \(A\in M_{m\times n}(\mathbb{K})\text{,}\) by applying a sequence of elementary row operations, can be reduced to a row reduced echelon form \(E_A\) of \(A\). The matrix \(E_A\) has the following properties.

All zero rows occurs at the bottom of \(E_A\text{.}\)
If a row of \(E_A\) is nonzero then the first nonzero entry is \(1\text{,}\) it is called pivot.
In any two successive nonzero rows of \(E_A\text{,}\) the pivot in the lower row occurs to the right of the pivot in the higher row.
Each column of \(E_A\) that contains the pivot has zero everywhere else.

The natural question arises: Does the sequence in which row operations are performed change the resulting row reduced echelon form of \(A?\) The answer is no!

Fact 1.3.4.

The row reduced echelon form of a matrix is unique.

Theorem 1.3.5.

Let \(\left(A^\prime|B^\prime\right)\) be a block row reduced echelon form of \(\left(A|B\right)\text{,}\) where \(B^\prime\) and \(B\) are column vectors. The system \(AX=B\) has a solution if and only if the system \(A^\prime X=B^\prime\) has a solution.
Let \(\left(A^\prime|B^\prime\right)\) be a block row reduced echelon form of \(\left(A|B\right)\text{,}\) where \(B^\prime\) and \(B\) are column vectors. The system \(A^\prime X=B^\prime\) has a solution if and only if there is no pivot in the last column \(B^\prime\text{.}\) In that case, one may assign arbitrary values to \(x_i\) if the \(i\)-th column does not contain a pivot.
Every system \(AX=0\) (here \(0\) is \(m\times 1\) matrix with all entries zero) of \(m\) equations and \(n\) unknowns with \(m<n\) has a nonzero solution.

Theorem 1.3.6.

A square matrix \(A\in M_n(\K)\) is invertible (see Definition 1.1.12) if and only if the row reduced echelon form of \(A\) is the identity matrix \(I_n\text{.}\)

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