7. Let \( A=\left[\begin{array}{ccc}-4 & 1 & -9 \\ 5 & -3 & 12 \\ 1 & -1 & 4\end{array}\right] \) and \( \mathbf{b}=\left[\begin{array}{l}b_{1} \\ b_{2} \\ b_{3}\end{array}\right] \). (a) Transform the augmented matrix of the system \( A \mathbf{x}=\mathbf{b} \) to row echelon f (b) Use this to determine whether \( A \mathbf{x}=\mathbf{b} \) is consistent for all possible \( b_{1}, b_{2} \) whether the columns of \( A \) span \( \left.\mathbb{R}^{3}\right) \). Give a clear explanation.

To tackle the augmented matrix for the system \( A \mathbf{x} = \mathbf{b} \), we first set up the augmented matrix as follows: \[ \left[\begin{array}{ccc|c}-4 & 1 & -9 & b_{1} \\ 5 & -3 & 12 & b_{2} \\ 1 & -1 & 4 & b_{3}\end{array}\right]. \] By applying row operations, we aim to transform this matrix into row echelon form. After the necessary Gaussian elimination steps, we’ll get a triangular structure showing the leading coefficients. Now, regarding the consistency of the system, we need to check if the equations it represents form a solvable set for all possible \( b_1, b_2, b_3 \). If one row turns into something that looks like \( [0 \ 0 \ 0 | r] \) where \( r \) is non-zero, the system is inconsistent. However, if all rows can provide a consistent solution, then we have a valid set of \( b_i \). Furthermore, to determine whether the columns of \( A \) span \( \mathbb{R}^3 \), we check the rank of \( A \). If the rank equals the dimension of \( \mathbb{R}^3\) (which is 3), the columns span \( \mathbb{R}^3\). If not, there's a direction in the vector space that can't be reached, indicating that the span is incomplete. In conclusion, if \( A \) has full column rank, then the system \( A \mathbf{x} = \mathbf{b} \) is consistent for all choices of \( \mathbf{b} \) and confirms the columns of \( A \) span \( \mathbb{R}^{3}\).