Question-2
Identify which of the following are in row echelon form and which are in reduced row echelon form. Also identify the pivots along the way.
\[ \begin{aligned} A & =\begin{bmatrix}0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix} & B & =\begin{bmatrix}-1 & 2 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{bmatrix} & C & =\begin{bmatrix}1 & 2 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{bmatrix}\\ \\D & =\begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} & E & =\begin{bmatrix}0 & 1 & 3\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{bmatrix} & F & =\begin{bmatrix}1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\\ \\G & =\begin{bmatrix}0 & 1 & 3 & -1\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 \end{bmatrix} & H & =\begin{bmatrix}0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix} & I & =\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\\ \\J & =\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0 \end{bmatrix} & K & =\begin{bmatrix}1 & 0 & 0 & -1\\ 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} & L & =\begin{bmatrix}1 & 0 & -1\\ 0 & 1 & 1 \end{bmatrix} \end{aligned} \]
Pivot
The pivot is the first non-zero entry in a row. As an example, the numbers marked in bold are the pivots:
\[ \begin{bmatrix}\boldsymbol{1} & 0 & -1 & 2\\ 0 & \boldsymbol{2} & 1 & 0\\ 0 & 0 & 0 & \boldsymbol{-1}\\ 0 & 0 & 0 & 0 \end{bmatrix} \]
Each non-zero row has a pivot. A zero row doesn’t have a pivot. The pivot is also called the leading entry.
REF
For a matrix to be in row echelon form, the following conditions have to be satisfied:
The pivot in any row should be to the right of the pivot in the previous row.
The pivot is \(1\). [this condition is not binding according to some authors, but it is binding for this course]
All zero rows should come at the end.
RREF
For a matrix to be in reduced row echelon form, the following conditions have to be satisfied. A column that contains a pivot is called a pivot column:
The matrix should be in row echelon form.
The pivot should be the only non-zero entry in a pivot column.
Solutions
\[ A=\begin{bmatrix}0 & \boldsymbol{1} & 0\\ \boldsymbol{1} & 0 & 1\\ 0 & \boldsymbol{1} & 0 \end{bmatrix} \]
\(A\) is not in REF. The pivot in the second row is to the left of the pivot in the first row.
\[ B=\begin{bmatrix}\boldsymbol{-1} & 2 & 0\\ 0 & 0 & \boldsymbol{1}\\ 0 & 0 & 0 \end{bmatrix} \]
\(B\) is not in REF. The pivot in the first row is \(-1\).
\[ C=\begin{bmatrix}\boldsymbol{1} & 2 & 0\\ 0 & 0 & \boldsymbol{1}\\ 0 & 0 & 0 \end{bmatrix}\] \(C\) is in RREF. The first and last columns are pivot columns.
\[ D=\begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \]
\(D\) is in RREF. It doesn’t have any pivots though.
\[ E=\begin{bmatrix}0 & \boldsymbol{1} & 3\\ 0 & \boldsymbol{1} & 1\\ 0 & 0 & \boldsymbol{1} \end{bmatrix}\] \(E\) is not in REF. The pivot in the second row is below the pivot in the first row.
\[ F=\begin{bmatrix}\boldsymbol{1} & 1 & 0\\ 0 & \boldsymbol{1} & 0\\ 0 & 0 & \boldsymbol{1} \end{bmatrix} \]
\(F\) is in REF. It is not in RREF as the second pivot column has two non-zero entries.
\[ G=\begin{bmatrix}0 & \boldsymbol{1} & 3 & -1\\ 0 & 0 & \boldsymbol{1} & 1\\ 0 & 0 & 0 & \boldsymbol{1} \end{bmatrix} \]
\(G\) is in REF. It is not in RREF since the third and fourth pivot columns have other non-zero entries.
\[ H=\begin{bmatrix}0 & 0 & 0\\ \boldsymbol{1} & 0 & 0\\ 0 & \boldsymbol{1} & 0\\ 0 & 0 & 0 \end{bmatrix} \]
\(H\) is not in REF. The first row is a zero row and for a matrix to be in REF, all zero rows should come at the end.
\[ I=\begin{bmatrix}\boldsymbol{1} & 0 & 0\\ 0 & \boldsymbol{1} & 0\\ 0 & 0 & \boldsymbol{1} \end{bmatrix}\] \(I\) is in RREF.
\[ J=\begin{bmatrix}0 & \boldsymbol{1} & 0\\ 0 & 0 & \boldsymbol{1}\\ 0 & 0 & 0\\ \boldsymbol{1} & 0 & 0 \end{bmatrix} \]
\(J\) is not in REF. There is a zero row above a non-zero row.
\[ K=\begin{bmatrix}\boldsymbol{1} & 0 & 0 & -1\\ 0 & \boldsymbol{1} & 1 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \]
\(K\) is in RREF. Do not be misled by the \(-1\) that appears at the end of the first row or the \(1\) that appears after the pivot in the second row.
\[ L=\begin{bmatrix}\boldsymbol{1} & 0 & -1\\ 0 & \boldsymbol{1} & 1 \end{bmatrix} \]
\(L\) is in RREF.