Answer to Question #196871 in Linear Algebra for Mpopo

Question

Determine whether or not the following matrices are in row echelon form or not? (4.1)

(1 2 −2)

( 0 1 2)

( 0 0 1)

(4.2) (1 2 −2)

(0 1 2 )

( 0 0 1 ) 

Answer 1

A matrix is in row echelon form (ref) when it satisfies the following conditions.

The first non-zero element in each row, called the leading entry, is 1.

Each leading entry is in a column to the right of the leading entry in the previous row.

Rows with all zero elements, if any, are below rows having a non-zero element.

4.1 Yes. The matrix

$$\begin{pmatrix} 1 & 2 &-2 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

is in row echelon form, since matrix satisfies all of the requirements for a row echelon matrix.

But the matrix is not in reduced row echelon matrix.

$R_1=R_1-2R_2$

$$\begin{pmatrix} 1 & 0 & -6 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

$R_1=R_1+6R_3$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

$R_2=R_2-2R_3$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

4.2 Yes. The matrix

$$\begin{pmatrix} 1 & 2 &-2 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

is in row echelon form, since matrix satisfies all of the requirements for a row echelon matrix.

But the matrix is not in reduced row echelon matrix.

$R_1=R_1-2R_2$

$$\begin{pmatrix} 1 & 0 & -6 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

$R_1=R_1+6R_3$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$

$R_2=R_2-2R_3$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$