Chapter 2

Rectangular Systems and Echelon Forms

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Section 2.1Row Echelon Form and Rank[6/100%]
Problem 2.1.1

Reduce each of the following matrices to row echelon form, determine the rank, and identify the basic columns.

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(a)

[123324692676]\begin{bmatrix} 1 & 2 & 3 & 3 \\ 2 & 4 & 6 & 9 \\ 2 & 6 & 7 & 6 \end{bmatrix}

(b)

[123268260125386]\begin{bmatrix} 1 & 2 & 3 \\ 2 & 6 & 8 \\ 2 & 6 & 0 \\ 1 & 2 & 5 \\ 3 & 8 & 6 \end{bmatrix}

(c)

[2113041424415521310436348195003−3003842141133]\begin{bmatrix} 2 & 1 & 1 & 3 & 0 & 4 & 1 \\ 4 & 2 & 4 & 4 & 1 & 5 & 5 \\ 2 & 1 & 3 & 1 & 0 & 4 & 3 \\ 6 & 3 & 4 & 8 & 1 & 9 & 5 \\ 0 & 0 & 3 & -3 & 0 & 0 & 3 \\ 8 & 4 & 2 & 14 & 1 & 13 & 3 \end{bmatrix}

Problem 2.1.2

Determine which of the following matrices are in row echelon form:

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(a)

[123004010]\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 4 \\ 0 & 1 & 0 \end{bmatrix}

(b)

[000001000001]\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

(c)

[223−4007−8000−1]\begin{bmatrix} 2 & 2 & 3 & -4 \\ 0 & 0 & 7 & -8 \\ 0 & 0 & 0 & -1 \end{bmatrix}

(d)

[120010000100000001000000]\begin{bmatrix} 1 & 2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}

Problem 2.1.3

Suppose that AA is an m×nm \times n matrix. Give a short explanation of why each of the following statements is true.

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(a)

rk⁡(A)≤min⁡{m,n}\operatorname {rk}\left(A\right) \leq \min \{ m, n\}.

(b)

rk⁡(A)<m\operatorname {rk}\left(A\right) < m if one row in AA is entirely zero.

(c)

rk⁡(A)<m\operatorname {rk}\left(A\right) < m if one row in AA is a multiple of another row.

(d)

rk⁡(A)<m\operatorname {rk}\left(A\right) < m if one row in AA is a combination of other rows.

(e)

rk⁡(A)<n\operatorname {rk}\left(A\right) < n if one column in AA is entirely zero.

Problem 2.1.4

Let A=[.1.2.3.4.5.6.7.8.901]A = \begin{bmatrix} .1 & .2 & .3 \\ .4 & .5 & .6 \\ .7 & .8 & .901 \end{bmatrix}.

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(a)

Use exact arithmetic to determine rk⁡(A)\operatorname {rk}\left(A\right).

(b)

Now use 3-digit floating-point arithmetic (without partial pivoting or scaling) to determine rk⁡(A)\operatorname {rk}\left(A\right). This number might be called the ``3-digit numerical rank.''

(c)

What happens if partial pivoting is incorporated?

Problem 2.1.5

How many different ``forms'' are possible for a 3×43 \times 4 matrix that is in row echelon form?

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Problem 2.1.6

Suppose that [[c∣c]Ab]\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix} is reduced to a matrix [[c∣c]Ec]\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix}.

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(a)

Is [[c∣c]Ec]\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix} in row echelon form if EE is?

(b)

If [[c∣c]Ec]\begin{bmatrix} [c|c] E & \mathbf{c} \end{bmatrix} is in row echelon form, must EE be in row echelon form?

Section 2.2Reduced Row Echelon Form[5/100%]
Problem 2.2.1

Determine the reduced row echelon form for each of the following matrices and then express each nonbasic column in terms of the basic columns:

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(a)

[123324692676]\begin{bmatrix} 1 & 2 & 3 & 3 \\ 2 & 4 & 6 & 9 \\ 2 & 6 & 7 & 6 \end{bmatrix}

(b)

[2113041424415521310436348195003−3003842141133]\begin{bmatrix} 2 & 1 & 1 & 3 & 0 & 4 & 1 \\ 4 & 2 & 4 & 4 & 1 & 5 & 5 \\ 2 & 1 & 3 & 1 & 0 & 4 & 3 \\ 6 & 3 & 4 & 8 & 1 & 9 & 5 \\ 0 & 0 & 3 & -3 & 0 & 0 & 3 \\ 8 & 4 & 2 & 14 & 1 & 13 & 3 \end{bmatrix}

Problem 2.2.2

Construct a matrix AA whose reduced row echelon form is

EA=[120−3000001−40100000100000000100000000000000]. E_{A} = \begin{bmatrix} 1 & 2 & 0 & -3 & 0 & 0 & 0 \\ 0 & 0 & 1 & -4 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}.

Is AA unique?

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Problem 2.2.3

Suppose that AA is an m×nm \times n matrix. Give a short explanation of why rk⁡(A)<n\operatorname {rk}\left(A\right) < n whenever one column in AA is a combination of other columns in AA.

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Problem 2.2.4

Consider the following matrix:

A=[.1.2.3.4.5.6.7.8.901]. A = \begin{bmatrix} .1 & .2 & .3 \\ .4 & .5 & .6 \\ .7 & .8 & .901 \end{bmatrix}.
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(a)

Use exact arithmetic to determine EAE_{A}.

(b)

Now use 3-digit floating-point arithmetic (without partial pivoting or scaling) to determine EAE_{A} and formulate a statement concerning ``near relationships'' between the columns of AA.

Problem 2.2.5

Consider the matrix

E=[10−1012000]. E = \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}.

You already know that E∗3E_{*3} can be expressed in terms of E∗1E_{*1} and E∗2E_{*2}. However, this is not the only way to represent the column dependencies in EE. Show how to write E∗1E_{*1} in terms of E∗2E_{*2} and E∗3E_{*3} and then express E∗2E_{*2} as a combination of E∗1E_{*1} and E∗3E_{*3}.

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Remark.
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This exercise illustrates that the set of pivotal columns is not the only set that can play the role of ``basic columns.'' Taking the basic columns to be the ones containing the pivots is a matter of convenience because everything becomes automatic that way.

Section 2.3Consistency of Linear Systems[8/100%]
Problem 2.3.1

Determine which of the following systems are consistent.

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(a)

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

(b)

[t]2x+2y+4z=0,3x+2y+5z=0,4x+2y+6z=0.\begin{aligned} [t] 2x + 2y + 4z & = 0, \\ 3x + 2y + 5z & = 0, \\ 4x + 2y + 6z & = 0. \end{aligned}

(c)

[t]x−y+z=1,x−y−z=2,x+y−z=3,x+y+z=4.\begin{aligned} [t] x - y + z & = 1, \\ x - y - z & = 2, \\ x + y - z & = 3, \\ x + y + z & = 4. \end{aligned}

(d)

[t]x−y+z=1,x−y−z=2,x+y−z=3,x+y+z=2.\begin{aligned} [t] x - y + z & = 1, \\ x - y - z & = 2, \\ x + y - z & = 3, \\ x + y + z & = 2. \end{aligned}

(e)

[t]2w+x+3y+5z=1,4w+4y+8z=0,w+x+2y+3z=0,x+y+z=0.\begin{aligned} [t] 2w + x + 3y + 5z & = 1, \\ 4w + 4y + 8z & = 0, \\ w + x + 2y + 3z & = 0, \\ x + y + z & = 0. \end{aligned}

(f)

[t]2w+x+3y+5z=7,4w+4y+8z=8,w+x+2y+3z=5,x+y+z=3.\begin{aligned} [t] 2w + x + 3y + 5z & = 7, \\ 4w + 4y + 8z & = 8, \\ w + x + 2y + 3z & = 5, \\ x + y + z & = 3. \end{aligned}

Problem 2.3.2

Construct a 3×43 \times 4 matrix AA and 3×13 \times 1 columns b\mathbf{b} and c\mathbf{c} such that [[c∣c]Ab]\begin{bmatrix} [c|c] A & \mathbf{b} \end{bmatrix} is the augmented matrix for an inconsistent system, but [[c∣c]Ac]\begin{bmatrix} [c|c] A & \mathbf{c} \end{bmatrix} is the augmented matrix for a consistent system.

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Problem 2.3.3

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

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Problem 2.3.4

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

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Problem 2.3.5

Is it possible for a parabola whose equation has the form y=α+βx+γx2y = \alpha + \beta x + \gamma x^{2} to pass through the four points (0,1)(0,1), (1,3)(1,3), (2,15)(2,15), and (3,37)(3,37)? Why?

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Problem 2.3.6

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

.835x+.667y=.168,.333x+.266y=.067. \begin{aligned} .835 x + .667 y & = .168, \\ .333 x + .266 y & = .067. \end{aligned}
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(a)

Is the system consistent when 5-digit arithmetic is used?

(b)

What happens when 6-digit arithmetic is used?

Problem 2.3.7

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 X\mathcal{X}, brand Y\mathcal{Y}, and brand Z\mathcal{Z}. One pound of brand X\mathcal{X} contains 2 units of phosphorous, 3 units of potassium, and 5 units of nitrogen. One pound of brand Y\mathcal{Y} contains 1 unit of phosphorous, 3 units of potassium, and 4 units of nitrogen. One pound of brand Z\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.

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Problem 2.3.8

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

(00⋯0∣α),α≠0 \begin{pmatrix} 0 & 0 & \cdots & 0 & \mid & \alpha \end{pmatrix}, \qquad \alpha \neq 0

does not appear in [[c∣c]Ec]\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?

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Section 2.4Homogeneous Systems[8/100%]
Problem 2.4.1

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

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(a)

[t]x1+2x2+x3+2x4=0,2x1+4x2+x3+3x4=0,3x1+6x2+x3+4x4=0.\begin{aligned} [t] x_{1} + 2 x_{2} + x_{3} + 2 x_{4} & = 0, \\ 2 x_{1} + 4 x_{2} + x_{3} + 3 x_{4} & = 0, \\ 3 x_{1} + 6 x_{2} + x_{3} + 4 x_{4} & = 0. \end{aligned}

(b)

[t]2x+y+z=0,4x+2y+z=0,6x+3y+z=0,8x+4y+z=0.\begin{aligned} [t] 2x + y + z & = 0, \\ 4x + 2y + z & = 0, \\ 6x + 3y + z & = 0, \\ 8x + 4y + z & = 0. \end{aligned}

(c)

[t]x1+x2+2x3=0,3x1+3x3+3x4=0,2x1+x2+3x3+x4=0,x1+2x2+3x3−x4=0.\begin{aligned} [t] x_{1} + x_{2} + 2 x_{3} & = 0, \\ 3 x_{1} + 3 x_{3} + 3 x_{4} & = 0, \\ 2 x_{1} + x_{2} + 3 x_{3} + x_{4} & = 0, \\ x_{1} + 2 x_{2} + 3 x_{3} - x_{4} & = 0. \end{aligned}

(d)

[t]2x+y+z=0,4x+2y+z=0,6x+3y+z=0,8x+5y+z=0.\begin{aligned} [t] 2x + y + z & = 0, \\ 4x + 2y + z & = 0, \\ 6x + 3y + z & = 0, \\ 8x + 5y + z & = 0. \end{aligned}

Problem 2.4.2

Among all solutions that satisfy the homogeneous system

x+2y+z=0,2x+4y+z=0,x+2y−z=0, \begin{aligned} x + 2y + z & = 0, \\ 2x + 4y + z & = 0, \\ x + 2y - z & = 0, \end{aligned}

determine those that also satisfy the nonlinear constraint y−xy=2zy - xy = 2z.

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Problem 2.4.3

Consider a homogeneous system whose coefficient matrix is

A=[1213124−13812357242623617−3]. A = \begin{bmatrix} 1 & 2 & 1 & 3 & 1 \\ 2 & 4 & -1 & 3 & 8 \\ 1 & 2 & 3 & 5 & 7 \\ 2 & 4 & 2 & 6 & 2 \\ 3 & 6 & 1 & 7 & -3 \end{bmatrix}.

First transform AA to an unreduced row echelon form to determine the general solution of the associated homogeneous system. Then reduce AA to EAE_{A}, and show that the same general solution is produced.

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Problem 2.4.4

If AA is the coefficient matrix for a homogeneous system consisting of four equations in eight unknowns and if there are five free variables, what is rk⁡(A)\operatorname {rk}\left(A\right)?

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Problem 2.4.5

Suppose that AA is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that AA has at least one nonzero row.

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(a)

Determine the fewest number of free variables that are possible.

(b)

Determine the maximum number of free variables that are possible.

Problem 2.4.6

Explain why a homogeneous system of mm equations in nn unknowns where m<nm < n must always possess an infinite number of solutions.

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Problem 2.4.7

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

x2[−2100]+x4[−3021] x_{2} \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_{4} \begin{bmatrix} -3 \\ 0 \\ 2 \\ 1 \end{bmatrix}

as its general solution.

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Problem 2.4.8

If c1\mathbf{c}_{1} and c2\mathbf{c}_{2} are columns that represent two particular solutions of the same homogeneous system, explain why the sum c1+c2\mathbf{c}_{1} + \mathbf{c}_{2} must also represent a solution of this system.

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Section 2.5Nonhomogeneous Systems[8/100%]
Problem 2.5.1

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

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(a)

[t]x1+2x2+x3+2x4=3,2x1+4x2+x3+3x4=4,3x1+6x2+x3+4x4=5.\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)

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

(c)

[t]x1+x2+2x3=1,3x1+3x3+3x4=6,2x1+x2+3x3+x4=3,x1+2x2+3x3−x4=0.\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)

[t]2x+y+z=2,4x+2y+z=5,6x+3y+z=8,8x+5y+z=8.\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

x1+x2+2x3+2x4+x5=1,2x1+2x2+4x3+4x4+3x5=1,2x1+2x2+4x3+4x4+2x5=2,3x1+5x2+8x3+6x4+5x5=3, \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:

(x1−x2)2−4x52=0,x32−x52=0. \begin{aligned} (x_{1} - x_{2})^{2} - 4 x_{5}^{2} & = 0, \\ x_{3}^{2} - x_{5}^{2} & = 0. \end{aligned}
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Problem 2.5.3

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 X\mathcal{X}, brand Y\mathcal{Y}, and brand Z\mathcal{Z}. One pound of brand X\mathcal{X} contains 2 units of phosphorous, 3 units of potassium, and 5 units of nitrogen. One pound of brand Y\mathcal{Y} contains 1 unit of phosphorous, 3 units of potassium, and 4 units of nitrogen. One pound of brand Z\mathcal{Z} contains only 1 unit of phosphorous and 1 unit of nitrogen.

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(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 X\mathcal{X} costs $1 per pound, brand Y\mathcal{Y} costs $6 per pound, and brand Z\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

Consider the following system:

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

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

(b)

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

(c)

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

Problem 2.5.5

If columns s1\mathbf{s}_{1} and s2\mathbf{s}_{2} are particular solutions of the same nonhomogeneous system, must it be the case that the sum s1+s2\mathbf{s}_{1} + \mathbf{s}_{2} is also a solution?

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Problem 2.5.6

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

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Problem 2.5.7

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

[1010]+x2[−2100]+x4[−3021] \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.

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Problem 2.5.8

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

[[ccc∣c].835.667.5.168.333.266.1994.0671.671.3341.1.436]. \begin{bmatrix} [ccc|c] .835 & .667 & .5 & .168 \\ .333 & .266 & .1994 & .067 \\ 1.67 & 1.334 & 1.1 & .436 \end{bmatrix}.
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(a)

Determine the 4-digit general solution.

(b)

Determine the 5-digit general solution.

(c)

Determine the 6-digit general solution.

Section 2.6Electrical Circuits[4/100%]
Problem 2.6.1

Suppose that Ri=iR_{i} = i ohms and Ei=iE_{i} = i volts in the circuit shown in Figure 2.6.1.

Figure 2.6.1Figure 2.6.1

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(a)

Determine the six indicated currents.

(b)

Select node number 1 to use as a reference point and fix its potential to be 0 volts. With respect to this reference, calculate the potentials at the other three nodes. Check your answer by verifying the loop rule for each loop in the circuit.

Problem 2.6.2

Determine the three currents indicated in the following circuit: two 12-volt and 9-volt batteries sit in the two vertical middle branches of a circuit with three parallel vertical branches joined by two horizontal branches (top and bottom). The left vertical branch contains a 12-volt battery and a 1 Ω1\, \Omega resistor; the right vertical branch contains a 9-volt battery and a 1 Ω1\, \Omega resistor; the two branches are joined at the top by a path through a 5 Ω5\, \Omega resistor (carrying current I2I_{2}, directed from right to left) and through an 8 Ω8\, \Omega resistor (carrying current I1I_{1}, directed from left to right), meeting at a shared middle-top node between the two resistors, and are joined at the bottom by a single path through a 10 Ω10\, \Omega resistor (carrying current I3I_{3}, directed from left to right).

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Problem 2.6.3

Determine the two unknown EMFs in the following circuit: three parallel horizontal branches connect a common left node to a common right node. The top branch contains a 20-volt battery and a 6 Ω6\, \Omega resistor, and carries a current of 1 amp (directed downward into the left node from the branch above it, i.e., the 1 amp enters the left node from the top branch). The middle branch contains a 4 Ω4\, \Omega resistor and an unknown EMF E1E_{1}. The bottom branch contains an unknown EMF E2E_{2} and a 2 Ω2\, \Omega resistor, and carries a current of 2 amps (directed downward out of the left node into the bottom branch).

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Problem 2.6.4

Consider the circuit shown below and answer the following questions.

Wheatstone-bridge-style circuit for Problem 2.6.4Wheatstone-bridge-style circuit for Problem 2.6.4

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(a)

How many nodes does the circuit contain?

(b)

How many branches does the circuit contain?

(c)

Determine the total number of loops and then determine the number of simple loops.

(d)

Demonstrate that the simple loop equations form an ``independent'' system of equations in the sense that there are no redundant equations.

(e)

Verify that any three of the nodal equations constitute an ``independent'' system of equations.

(f)

Verify that the loop equation associated with the loop containing R1R_{1}, R2R_{2}, R3R_{3}, and R4R_{4} can be expressed as the sum of the two equations associated with the two simple loops contained in the larger loop.

(g)

Determine the indicated current II if R1=R2=R3=R4=1R_{1} = R_{2} = R_{3} = R_{4} = 1 ohm, R5=R6=5R_{5} = R_{6} = 5 ohms, and E=5E = 5 volts.