1st year University, reduced row echelon form

Question

Can someone explain me how each of these are incorrect answers

Thank you in advance.

Answer 1

For a matrix to be in Row Echelon form, it must satisfy the following properties: $$ \begin{aligned} &\bullet \text{ All nonzero rows are above any rows of all zeros.}\\ &\bullet \text{ Each leading entry of a row is in a column to the right of the leading entry of the row above it.}\\ &\bullet \text{ All entries in a column below a leading entry are zeros.}\end{aligned}$$ And, to be in Row Reduced Echelon Form, it must satisfy the previous properties with the following: $$ \begin{aligned} &\bullet \text{ The leading entry in each nonzero row is 1.}\\ &\bullet \text{ Each leading 1 is the only nonzero entry in its column.} \end{aligned} $$ Examle: $$1.\, \left( \begin{array}{ccccc} 1 & 1 & 0 & 1 & 1 \\ 0 & 2 & 0 & 2 & 2 \\ 0 & 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 & 4 \\ \end{array} \right)\qquad \qquad 2.\, \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)$$ $1$ is in Row Echelon Form, while $2$ is in Row Reduced Echelon Form.

Now applying the above properties to your question you can see that:

$$\begin{aligned} &-(a)\, \text{is not in RREF because there is a 5 above the pivot on the second row} \\ &-(b)\, \text{is not in RREF because there is a 5 above the pivot on the third row} \\ &-(c)\, \text{is not in RREF because there is a 1 above the pivot on the fourth row} \\ &-(e)\, \text{if you swap the last row with the second row, then it is in RREF} \\ \end{aligned}$$ This concludes that $(d)\, $ is the only matrix in $\text{RREF}$.