###### Preview Activity2.3.1The existence of solutions
1. If the equation $$A\xvec = \bvec$$ is inconsistent, what can we say about the pivots of the augmented matrix $$\left[\begin{array}{r|r} A \amp \bvec \end{array}\right]\text{?}$$

2. Consider the matrix $$A$$

\begin{equation*} A = \left[ \begin{array}{rrr} 1 \amp 0 \amp -2 \\ -2 \amp 2 \amp 2 \\ 1 \amp 1 \amp -3 \end{array}\right]\text{.} \end{equation*}

If $$\bvec=\threevec{2}{2}{5}\text{,}$$ is the equation $$A\xvec = \bvec$$ consistent? If so, find a solution.

3. If $$\bvec=\threevec{2}{2}{6}\text{,}$$ is the equation $$A\xvec = \bvec$$ consistent? If so, find a solution.

4. Identify the pivot positions of $$A\text{.}$$

5. For our two choices of the vector $$\bvec\text{,}$$ one equation $$A\xvec = \bvec$$ has a solution and the other does not. What feature of the pivot positions of the matrix $$A$$ tells us to expect this?

in-context