QuestionDo the columns of $A$ span $\mathbb{R}^{4}$? Does $A \mathbf{x}=\mathbf{b}$ have a solution for all $\mathbf{b}$?
$$A=\begin{bmatrix} 1 & 3 & 5 & -9 \\ 2 & 0 & -8 & 6 \\ 3 & 4 & 0 & -7 \\ -2 & -7 & -13 & 18 \end{bmatrix}$$

Studdy Solution
Since the columns of $A$ do not span $\mathbb{R}^{}$, the equation $A\mathbf{x}=\mathbf{b}$ does not have a solution for each $\mathbf{b}$ in $\mathbb{R}^{}$. This is because there are vectors in $\mathbb{R}^{}$ that cannot be expressed as a linear combination of the columns of $A$.
The correct choice is A. No, because the reduced echelon form of $A$ is$$\left[\begin{array}{rrrr} &0 &0 &/3 \\ 0 & &0 &/3 \\ 0 &0 &0 &0 \\ 0 &0 & & -5/3\end{array}\right]$$