Activity 1.4.3

Here are the three augmented matrices in reduced row echelon form that we considered in the previous section.

$$\left[ \begin{array}{rrr|r} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \\ \end{array} \right]$$

$$\left[ \begin{array}{rrr|r} 1 & 0 & 2 & 3 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right]$$

$$\left[ \begin{array}{rrr|r} 1 & 0 & 2 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} \right]$$

For each matrix, identify the pivot positions and state whether the corresponding linear system is consistent. If the system is consistent, explain whether the solution is unique or whether there are infinitely many solutions.

If possible, give an example of a $3\times5$ augmented matrix that corresponds to a linear system having a unique solution. If it is not possible, explain why.

If possible, give an example of a $5\times3$ augmented matrix that corresponds to a linear system having a unique solution. If it is not possible, explain why.

What condition on the pivot positions guarantees that a linear system has a unique solution?

If a linear system has a unique solution, what can we say about the relationship between the number of equations and the number of variables?