Row Echelon Form and Rank

$\begin{bmatrix} 1 & 2 & 3 & 3 \\ 2 & 4 & 6 & 9 \\ 2 & 6 & 7 & 6 \end{bmatrix}$

$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 6 & 8 \\ 2 & 6 & 0 \\ 1 & 2 & 5 \\ 3 & 8 & 6 \end{bmatrix}$

$\begin{bmatrix} 2 & 1 & 1 & 3 & 0 & 4 & 1 \\ 4 & 2 & 4 & 4 & 1 & 5 & 5 \\ 2 & 1 & 3 & 1 & 0 & 4 & 3 \\ 6 & 3 & 4 & 8 & 1 & 9 & 5 \\ 0 & 0 & 3 & -3 & 0 & 0 & 3 \\ 8 & 4 & 2 & 14 & 1 & 13 & 3 \end{bmatrix}$

$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 4 \\ 0 & 1 & 0 \end{bmatrix}$

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

$\begin{bmatrix} 2 & 2 & 3 & -4 \\ 0 & 0 & 7 & -8 \\ 0 & 0 & 0 & -1 \end{bmatrix}$

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

$\operatorname {rk}\left(A\right) \leq \min \{ m, n\}$.

$\operatorname {rk}\left(A\right) < m$ if one row in $A$ is entirely zero.

$\operatorname {rk}\left(A\right) < m$ if one row in $A$ is a multiple of another row.

$\operatorname {rk}\left(A\right) < m$ if one row in $A$ is a combination of other rows.

$\operatorname {rk}\left(A\right) < n$ if one column in $A$ is entirely zero.

Use exact arithmetic to determine $\operatorname {rk}\left(A\right)$.

Now use 3-digit floating-point arithmetic (without partial pivoting or scaling) to determine $\operatorname {rk}\left(A\right)$. This number might be called the ``3-digit numerical rank.''

What happens if partial pivoting is incorporated?

Is $\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix}$ in row echelon form if $E$ is?

If $\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix}$ is in row echelon form, must $E$ be in row echelon form?