Preview Activity 1.4.1 Some basic observations about pivots

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*} How many pivot positions can there be in one row? In a \(3\times5\) 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.
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.
Give an example of a matrix with a pivot position in every row and every column. What is special about such a matrix?