Consistency of Linear Systems

Determine which of the following systems are consistent.

Construct a $3 \times 4$ matrix $A$ and $3 \times 1$ columns $\mathbf{b}$ and $\mathbf{c}$ such that $\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix}$ is the augmented matrix for an inconsistent system, but $\begin{bmatrix} [c|c] A & \mathbf{c} \end{bmatrix}$ is the augmented matrix for a consistent system.

If $A$ is an $m \times n$ matrix with $\operatorname {rk}\left(A\right) = m$, explain why the system $\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix}$ must be consistent for every right-hand side $\mathbf{b}$.

Consider two consistent systems whose augmented matrices are of the form $\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix}$ and $\begin{bmatrix} [c|c] A & \mathbf{c} \end{bmatrix}$. That is, they differ only on the right-hand side. Is the system associated with $\begin{bmatrix} [c|c] A & \mathbf{b}+\mathbf{c} \end{bmatrix}$ also consistent? Explain why.

Is it possible for a parabola whose equation has the form $y = \alpha + \beta x + \gamma x^{2}$ to pass through the four points $(0,1)$, $(1,3)$, $(2,15)$, and $(3,37)$? Why?

Consider using floating-point arithmetic (without scaling) to solve the following system:

In order to grow a certain crop, it is recommended that each square foot of ground be treated with 10 units of phosphorous, 9 units of potassium, and 19 units of nitrogen. Suppose that there are three brands of fertilizer on the market -- say brand $\mathcal{X}$, brand $\mathcal{Y}$, and brand $\mathcal{Z}$. One pound of brand $\mathcal{X}$ contains 2 units of phosphorous, 3 units of potassium, and 5 units of nitrogen. One pound of brand $\mathcal{Y}$ contains 1 unit of phosphorous, 3 units of potassium, and 4 units of nitrogen. One pound of brand $\mathcal{Z}$ contains only 1 unit of phosphorous and 1 unit of nitrogen. Determine whether or not it is possible to meet exactly the recommendation by applying some combination of the three brands of fertilizer.

Suppose that an augmented matrix $\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix}$ is reduced by means of Gaussian elimination to a row echelon form $\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix}$. If a row of the form

does not appear in $\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix}$, is it possible that rows of this form could have appeared at earlier stages in the reduction process? Why?