Exercise6

Consider the matrix

\begin{equation*} A=\left[\begin{array}{rrrr} 1 \amp 2 \amp -4 \amp -4 \\ 2 \amp 3 \amp 0 \amp 1 \\ 1 \amp 0 \amp 4 \amp 6 \\ \end{array}\right] \text{.} \end{equation*}
  1. Find the product \(A\xvec\) where

    \begin{equation*} \xvec = \fourvec{1}{-2}{0}{2} \text{.} \end{equation*}
  2. Give a description of the vectors \(\xvec\) such that

    \begin{equation*} A\xvec = \threevec{-1}{15}{17} \text{.} \end{equation*}

  3. Find the reduced row echelon form of \(A\) and identify the pivot positions.

  4. Can you find a vector \(\bvec\) such that \(A\xvec=\bvec\) is inconsistent?

  5. For a general 3-dimensional vector \(\bvec\text{,}\) what can you say about the solution space of the equation \(A\xvec = \bvec\text{?}\)

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