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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 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 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 is Row Reduced Echelon form. 

 

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