Created almost 9 years ago, updated 5 months ago
ID23541
Examples of matrices in row echelon form:
-
$$\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
— Pivot positions: (1,1) and (2,3). Zero row at bottom.
-
$$\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix}$$
— Already in echelon form (also upper triangular).
-
$$\begin{pmatrix} 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
— First column has no pivot; pivots in columns 2 and 4.
Non-examples:
- $$\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$$ — entry below pivot is not zero
- $$\begin{pmatrix} 0 & 0 \\ 1 & 2 \end{pmatrix}$$ — zero row is above a nonzero row
**Examples of matrices in row echelon form:**
1. $$\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
— Pivot positions: (1,1) and (2,3). Zero row at bottom.
2. $$\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix}$$
— Already in echelon form (also upper triangular).
3. $$\begin{pmatrix} 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
— First column has no pivot; pivots in columns 2 and 4.
**Non-examples:**
- $$\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$$ — entry below pivot is not zero
- $$\begin{pmatrix} 0 & 0 \\ 1 & 2 \end{pmatrix}$$ — zero row is above a nonzero row