Question

Which of the following is in reduced row Echelon form

• A. $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$
• C. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Answer 1

Answer: (C), (D)

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Apart from being in Row echelon form if the matrix obeys this following condition it will be in reduced Row echelon form.

$\to$ Every leading entry which 1 should be the only non zero element in that column.

In $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$3^{rd}$ column has 2 entries which contradicts the extra condition.

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$ is not in reduced Row Echelon form. For it be in reduced row echelon we have to interchange $R_2\leftrightarrow R_3$.

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ This satisfied all the conditions of Reduced Row Echelon form.

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$ This too satisfied all the conditions for reduced Row Ehlon form.