Exercise 6

Consider the matrix $$A=\left[\begin{array}{rrrr} 1 & 2 & -4 & -4 \\ 2 & 3 & 0 & 1 \\ 1 & 0 & 4 & 6 \\ \end{array}\right]$$.

1. Find the product $A\xvec$ where $$\xvec = \fourvec{1}{-2}{0}{2}$$.

2. Give a description of the vectors $\xvec$ such that $$A\xvec = \threevec{-1}{15}{17}$$.

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 $\bvec$, what can you say about the solution space of the equation $A\xvec = \bvec$?