Echelon Matrix/Examples
Examples of Echelon Matrices
Arbitrary Example $1$
- $\begin {bmatrix}
1 & 0 & -1 & 2 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix} $ is an echelon matrix, but not a reduced echelon matrix, because the leading $1$ in row $3$ is not the only $1$ in its column.
Arbitrary Example $2$
- $\begin {bmatrix}
1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix} $ is a reduced echelon matrix.
Arbitrary Example $3$
- $\begin {bmatrix}
0 & 1 & 0 & 2 \\ 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix} $ is not an echelon matrix, because the leading $1$ in row $2$ is to the left of the leading $1$ in row $1$.
Arbitrary Example $4$
- $\begin {bmatrix}
1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ \end {bmatrix} $ is not an echelon matrix, because row $2$ is a zero row, coming before row $3$.
Arbitrary Example $5$
- $\begin {bmatrix}
1 & 5 & 4 & 2 \\ 0 & 6 & 0 & 9 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$ is not an echelon matrix, because the leading coefficient of row $2$ is not $1$.
It is, however, a non-unity variant of an echelon matrix.
Arbitrary Example $6$
- $\begin {bmatrix}
1 & 6 & -1 & 4 & 2 \\ 0 & 0 & 1 & 2 & -3 \\ 0 & 0 & 0 & 1 & 5 \\ \end {bmatrix}$ is an echelon matrix, but not a reduced echelon matrix, because the leading $1$ in row $3$ is not the only $1$ in its column.
Arbitrary Example $7$
- $\begin {bmatrix}
1 & 6 & -1 & 4 & 2 \\ 0 & 1 & 2 & -3 & 5 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$ is an echelon matrix, but not a reduced echelon matrix, because the leading $1$ in row $2$ is not the only $1$ in its column.
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Example