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 \(A\) tell us to expect this?

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