##### Preview Activity1.4.1Some basic observations about pivots

1. Given below is a matrix and its reduced row echelon form. Indicate the pivot positions.

\begin{equation*} \left[ \begin{array}{rrrr} 2 \amp 4 \amp 6 \amp -1 \\ -3 \amp 1 \amp 5 \amp 0 \\ 1 \amp 3 \amp 5 \amp 1 \\ \end{array} \right] \sim \left[ \begin{array}{rrrr} 1 \amp 0 \amp -1 \amp 0 \\ 0 \amp 1 \amp 2 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \\ \end{array} \right] \text{.} \end{equation*}
2. How many pivot positions can there be in one row? In an $$3\times5$$ matrix, what is the largest possible number of pivot positions? Give an example of a matrix that has the largest possible number of pivot positions.

3. How many pivots can there be in one column? In a $$5\times3$$ matrix, what is the largest possible number of pivot positions? Give an example of a matrix that has the largest possible number of pivot positions.

4. Give an example of a matrix with a pivot position in every row and every column. What is special about such a matrix?

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