Q1: How many pivots does the following row echelon form have? $$ \left( \begin{array}{cccc}1 & 0 & 3 & 0 \ 0 & 3 & 1 & 0 \ 0 & 0 & 1 & 1 \ \end{array} \right) $$

A1: $3$

A2: $1$

A3: $0$

A4: $2$

A5: $4$

Q2: Is the following matrix in row echelon form? $$ \left( \begin{array}{cccc}1 & 0 & 3 & 0 \ 0 & 3 & 1 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 1 \ \end{array} \right) $$

A1: No, the zero-row is not at the bottom of the matrix.

A2: No, we need 5 pivots.

A3: Yes and we have 3 pivots.

A4: Yes, for each row the first non-zero is strictly to the right of the first non-zero entry of the row above.

Q3: What are the free variables for this row echelon form? $$ \left( \begin{array}{cccc}1 & 0 & 3 & 0 \ 0 & 3 & 1 & 0 \ 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ \end{array} \right) $$

A1: There are none.

A2: All variables are free variables.

A3: $x_1, x_2$

A4: Only $x_1$

A5: Only $x_4$