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

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.
If \(\bvec=\threevec{2}{2}{6}\text{,}\) is the equation \(A\xvec = \bvec\) consistent? If so, find a solution.
Identify the pivot positions of \(A\text{.}\)
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 \(A\) tell us to expect this?