Row Reduced Echelon Form is a specific Echelon Form of a matrix that can be obtained by modifying a matrix with basic row operations, such that:
- Every leading coefficient is 1, and
- It is the only non-zero entry in its column
One example of Row Reduced Echelon form:
$\begin{bmatrix}
\color{red}1 & 0 & 0 & 0 \\
0 & 0 & \color{red}1 & 0 \\
0 & 0 & 0 & \color{red}1
\end{bmatrix}$
We have three pivot columns in above matrix and we can observe that each pivot is value $\color{red}1$ and only non-zero entry in its column.
Now, let’s go one by one for each option in the question below:
A. $\begin{bmatrix}
\color{red}1 & 0 & 0 & 0\\
0 & 0 & \color{red}1 & 0\\
0 & 0 & 0 & \color{red}1\\
0 & 0 & 0 & 0
\end{bmatrix}$
$=> $ Matrix satisfies the condition that pivot value is 1 only, and is the only non-zero entry in its column, so A is Row Reduced Echelon form
B. $\begin{bmatrix}
\color{red}1 & 0 & 0 & 0\\
0 & \color{red}1 & 0 & 0\\
0 & 0 & \color{red}1 & 0\\
0 & 0 & 0 & \color{red}1
\end{bmatrix}$
$=> $ Matrix satisfies the condition that pivot value is 1 only, and is the only non-zero entry in its column, so B is Row Reduced Echelon form. Even we can say that the matrix is so called Convenient/Identity Matrix.
C. $\begin{bmatrix}
0 & \color{red}1 & 0 & 0\\
0 & 0 & 0 & \color{red}1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}$
$=> $ Matrix satisfies the condition that pivot value is 1 only, and is the only non-zero entry in its column, so C is Row Reduced Echelon form.