Math Problems Vault

Problem 2.5.1

(a)(b)(c)(d)

Determine the general solution for each of the following nonhomogeneous systems.

?

(a)

$\begin{aligned} [t] x_{1} + 2 x_{2} + x_{3} + 2 x_{4} & = 3, \\ 2 x_{1} + 4 x_{2} + x_{3} + 3 x_{4} & = 4, \\ 3 x_{1} + 6 x_{2} + x_{3} + 4 x_{4} & = 5. \end{aligned}$

(b)

$\begin{aligned} [t] 2x + y + z & = 4, \\ 4x + 2y + z & = 6, \\ 6x + 3y + z & = 8, \\ 8x + 4y + z & = 10. \end{aligned}$

(c)

$\begin{aligned} [t] x_{1} + x_{2} + 2 x_{3} & = 1, \\ 3 x_{1} + 3 x_{3} + 3 x_{4} & = 6, \\ 2 x_{1} + x_{2} + 3 x_{3} + x_{4} & = 3, \\ x_{1} + 2 x_{2} + 3 x_{3} - x_{4} & = 0. \end{aligned}$

(d)

$\begin{aligned} [t] 2x + y + z & = 2, \\ 4x + 2y + z & = 5, \\ 6x + 3y + z & = 8, \\ 8x + 5y + z & = 8. \end{aligned}$

Problem 2.5.2

Among the solutions that satisfy the set of linear equations

\begin{aligned} x_{1} + x_{2} + 2 x_{3} + 2 x_{4} + x_{5} & = 1, \\ 2 x_{1} + 2 x_{2} + 4 x_{3} + 4 x_{4} + 3 x_{5} & = 1, \\ 2 x_{1} + 2 x_{2} + 4 x_{3} + 4 x_{4} + 2 x_{5} & = 2, \\ 3 x_{1} + 5 x_{2} + 8 x_{3} + 6 x_{4} + 5 x_{5} & = 3, \end{aligned}

find all those that also satisfy the following two constraints:

\begin{aligned} (x_{1} - x_{2})^{2} - 4 x_{5}^{2} & = 0, \\ x_{3}^{2} - x_{5}^{2} & = 0. \end{aligned}

?

Problem 2.5.3

(a)(b)

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 -- 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.

?

(a)

Take into account the obvious fact that a negative number of pounds of any brand can never be applied, and suppose that because of the way fertilizer is sold only an integral number of pounds of each brand will be applied. Under these constraints, determine all possible combinations of the three brands that can be applied to satisfy the recommendations exactly.

(b)

Suppose that brand $\mathcal{X}$ costs $1 per pound, brand $\mathcal{Y}$ costs $6 per pound, and brand $\mathcal{Z}$ costs $3 per pound. Determine the least expensive solution that will satisfy the recommendations exactly as well as the constraints of part (a).

Problem 2.5.4

(a)(b)(c)

Consider the following system:

\begin{aligned} 2x + 2y + 3z & = 0, \\ 4x + 8y + 12z & = -4, \\ 6x + 2y + \alpha z & = 4. \end{aligned}

?

(a)

Determine all values of $\alpha$ for which the system is consistent.

(b)

Determine all values of $\alpha$ for which there is a unique solution, and compute the solution for these cases.

(c)

Determine all values of $\alpha$ for which there are infinitely many different solutions, and give the general solution for these cases.

Problem 2.5.5

If columns $\mathbf{s}_{1}$ and $\mathbf{s}_{2}$ are particular solutions of the same nonhomogeneous system, must it be the case that the sum $\mathbf{s}_{1} + \mathbf{s}_{2}$ is also a solution?

?

Problem 2.5.6

Suppose that $\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix}$ is the augmented matrix for a consistent system of $m$ equations in $n$ unknowns where $m \geq n$ . What must $E_{A}$ look like when the system possesses a unique solution?

?

Problem 2.5.7

Construct a nonhomogeneous system of three equations in four unknowns that has

\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix} + x_{2} \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_{4} \begin{bmatrix} -3 \\ 0 \\ 2 \\ 1 \end{bmatrix}

as its general solution.

?

Problem 2.5.8

(a)(b)(c)

Consider using floating-point arithmetic (without partial pivoting or scaling) to solve the system represented by the following augmented matrix:

\begin{bmatrix} [ccc|c] .835 & .667 & .5 & .168 \\ .333 & .266 & .1994 & .067 \\ 1.67 & 1.334 & 1.1 & .436 \end{bmatrix}.

?

(a)

Determine the 4-digit general solution.

(b)

Determine the 5-digit general solution.

(c)

Determine the 6-digit general solution.