Section 18

Systems of linear homogeneous equations

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Problem 18.1

Write out the coefficient matrix of the given system of homogeneous linear equations. Solve the system:

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(1)

x−y=0x - y = 0

(2)

x1−x2+2x3=0x_{1} - x_{2} + 2x_{3} = 0

(3)

x1+x2+x3+x4+x5=0x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 0

(4)

x+3y+2z=0x + 3y + 2z = 0, 2x+4y+3z=02x + 4y + 3z = 0

(5)

5x−8y+3z=05x - 8y + 3z = 0, 2x−3y+z=02x - 3y + z = 0

(6)

x+2y+3z=0x + 2y + 3z = 0, 2x+3y+4z=02x + 3y + 4z = 0, x+y+z=0x + y + z = 0

(7)

5x1−8x2+3x3+3x4=05x_{1} - 8x_{2} + 3x_{3} + 3x_{4} = 0, 4x1−6x2+2x3+x4=04x_{1} - 6x_{2} + 2x_{3} + x_{4} = 0

(8)

x1+3x2+4x3+x4=0x_{1} + 3x_{2} + 4x_{3} + x_{4} = 0, 2x1+2x2+3x3+x4=02x_{1} + 2x_{2} + 3x_{3} + x_{4} = 0, 3x1+3x2+4x3+x4=03x_{1} + 3x_{2} + 4x_{3} + x_{4} = 0

(9)

2x1+4x2+6x3+x4=02x_{1} + 4x_{2} + 6x_{3} + x_{4} = 0, x1+2x2+3x3+x4=0x_{1} + 2x_{2} + 3x_{3} + x_{4} = 0, 3x1+6x2+9x3−x4=03x_{1} + 6x_{2} + 9x_{3} - x_{4} = 0, x1+2x2+3x3+5x4=0x_{1} + 2x_{2} + 3x_{3} + 5x_{4} = 0

(10)

x1+x2+x3−x4=0x_{1} + x_{2} + x_{3} - x_{4} = 0, 3x1+2x2+x3−x5=03x_{1} + 2x_{2} + x_{3} - x_{5} = 0

(11)

x1+3x2+3x3+2x4+6x5=0x_{1} + 3x_{2} + 3x_{3} + 2x_{4} + 6x_{5} = 0, x1−x2−2x3−3x5=0x_{1} - x_{2} - 2x_{3} - 3x_{5} = 0, x1+11x2+7x3+6x4+18x5=0x_{1} + 11x_{2} + 7x_{3} + 6x_{4} + 18x_{5} = 0

(12)

x1+3x2+2x3+2x4+5x5=0x_{1} + 3x_{2} + 2x_{3} + 2x_{4} + 5x_{5} = 0, 2x1+2x2+3x3+2x4+5x5=02x_{1} + 2x_{2} + 3x_{3} + 2x_{4} + 5x_{5} = 0, 3x1+x2+x3+2x4+2x5=03x_{1} + x_{2} + x_{3} + 2x_{4} + 2x_{5} = 0, x1+x2−x3+x4=0x_{1} + x_{2} - x_{3} + x_{4} = 0.

Problem 18.2

Prove that:

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(1)

the sum of two solutions of a homogeneous system of linear equations is a solution of the same system

(2)

the product of some solution of a homogeneous system of linear equations by a number is a solution of the same system.

Problem 18.3

Let kk be the maximal number of linearly independent solutions of a homogeneous system of linear equations. Express kk in terms of the size and rank of the matrix of the system. In what case is k=0k = 0?

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Problem 18.4

How many linearly independent solutions does a homogeneous system of linear equations have, if its matrix is nonsingular?

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Problem 18.5

Can a homogeneous system of linear equations turn out to be inconsistent?

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Problem 18.6

Formulate the conditions (and verify their necessity and sufficiency) under which a homogeneous system of linear equations has:

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(1)

a unique solution

(2)

infinitely many solutions.

Problem 18.7

Compose and solve a homogeneous system of linear equations given by its coefficient matrix:

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(1)

[12]\left[\begin{smallmatrix} 1 & 2 \end{smallmatrix}\right]

(2)

[111]\left[\begin{smallmatrix} 1 & 1 & 1 \end{smallmatrix}\right]

(3)

[1301]\left[\begin{smallmatrix} 1 & 3 & 0 & 1 \end{smallmatrix}\right];

(4)

A391\mathbf{A}_{391}. A391=[689016]\mathbf{A}_{391} = \left[\begin{smallmatrix} 6 & 8 & 9 \\ 0 & 1 & 6 \end{smallmatrix}\right].

(5)

A455\mathbf{A}_{455}. A455=[1234234534564567]\mathbf{A}_{455} = \left[\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \end{smallmatrix}\right].

(6)

A500\mathbf{A}_{500}. A500=[524−7−1−1−273]\mathbf{A}_{500} = \left[\begin{smallmatrix} 5 & 24 & -7 & -1 \\ -1 & -2 & 7 & 3 \end{smallmatrix}\right].

(7)

A514\mathbf{A}_{514}. A514=[2−3574−6232−3−11−15]\mathbf{A}_{514} = \left[\begin{smallmatrix} 2 & -3 & 5 & 7 \\ 4 & -6 & 2 & 3 \\ 2 & -3 & -11 & -15 \end{smallmatrix}\right].

(8)

A519\mathbf{A}_{519}. A519=[12313211231−122215520]\mathbf{A}_{519} = \left[\begin{smallmatrix} 1 & 2 & 3 & 1 \\ 3 & 2 & 1 & 1 \\ 2 & 3 & 1 & -1 \\ 2 & 2 & 2 & 1 \\ 5 & 5 & 2 & 0 \end{smallmatrix}\right].

(9)

A573\mathbf{A}_{573}. A573=[0011101001]\mathbf{A}_{573} = \left[\begin{smallmatrix} 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{smallmatrix}\right].

(10)

A582\mathbf{A}_{582}. A582=[33−3−341−111−221−1−11]\mathbf{A}_{582} = \left[\begin{smallmatrix} 3 & 3 & -3 & -3 & 4 \\ 1 & -1 & 1 & 1 & -2 \\ 2 & 1 & -1 & -1 & 1 \end{smallmatrix}\right].

(11)

A583\mathbf{A}_{583}. A583=[−2111−11−12−117−51−55]\mathbf{A}_{583} = \left[\begin{smallmatrix} -2 & 1 & 1 & 1 & -1 \\ 1 & -1 & 2 & -1 & 1 \\ 7 & -5 & 1 & -5 & 5 \end{smallmatrix}\right].

Problem 18.8

Compose a homogeneous system of linear equations from the given coefficient matrix containing a parameter. Solve the system for all possible values of the parameter:

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(1)

A=A221−λEA = \mathbf{A}_{221} - \lambda E. A221=[41−125−244−1]\mathbf{A}_{221} = \left[\begin{smallmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 4 & 4 & -1 \end{smallmatrix}\right].

(2)

A=A212−λEA = \mathbf{A}_{212} - \lambda E. A212=[4−1−121−24−4−1]\mathbf{A}_{212} = \left[\begin{smallmatrix} 4 & -1 & -1 \\ 2 & 1 & -2 \\ 4 & -4 & -1 \end{smallmatrix}\right].

(3)

A=A222−λEA = \mathbf{A}_{222} - \lambda E. A222=[111222−3−3−3]\mathbf{A}_{222} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{smallmatrix}\right].

(4)

A=A365A = \mathbf{A}_{365}. A365=[1111αα1α2α2]\mathbf{A}_{365} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha \\ 1 & \alpha^{2} & \alpha^{2} \end{smallmatrix}\right].

(5)

A=A213−λEA = \mathbf{A}_{213} - \lambda E. A213=[111222333]\mathbf{A}_{213} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{smallmatrix}\right].

(6)

A=A363A = \mathbf{A}_{363}. A363=[1111ωω21ω2ω]\mathbf{A}_{363} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & \omega & \omega^{2} \\ 1 & \omega^{2} & \omega \end{smallmatrix}\right].

Problem 18.9

Solve a homogeneous system of linear equations given by its coefficient matrix. Compose and solve the corresponding adjoint system:

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(1)

A114\mathbf{A}_{114}. A114=[355947]\mathbf{A}_{114} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \\ 4 & 7 \end{smallmatrix}\right].

(2)

A115\mathbf{A}_{115}. A115=[443311]\mathbf{A}_{115} = \left[\begin{smallmatrix} 4 & 4 \\ 3 & 3 \\ 1 & 1 \end{smallmatrix}\right].

(3)

A118\mathbf{A}_{118}. A118=[345723]\mathbf{A}_{118} = \left[\begin{smallmatrix} 3 & 4 \\ 5 & 7 \\ 2 & 3 \end{smallmatrix}\right].

(4)

A205\mathbf{A}_{205}. A205=[123234345]\mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right].

(5)

A209\mathbf{A}_{209}. A209=[1−3−1−27232−4]\mathbf{A}_{209} = \left[\begin{smallmatrix} 1 & -3 & -1 \\ -2 & 7 & 2 \\ 3 & 2 & -4 \end{smallmatrix}\right].

(6)

A368\mathbf{A}_{368}. A368=[1+i2i−211+i3i−311+i4i−41]\mathbf{A}_{368} = \left[\begin{smallmatrix} 1+i \sqrt{2} & i-\sqrt{2} & 1 \\ 1+i \sqrt{3} & i-\sqrt{3} & 1 \\ 1+i \sqrt{4} & i-\sqrt{4} & 1 \end{smallmatrix}\right].

(7)

A408\mathbf{A}_{408}. A408=[242−1−2−115381−2274]\mathbf{A}_{408} = \left[\begin{smallmatrix} 2 & 4 & 2 \\ -1 & -2 & -1 \\ 1 & 5 & 3 \\ 8 & 1 & -2 \\ 2 & 7 & 4 \end{smallmatrix}\right].

(8)

A145\mathbf{A}_{145}. A145=[12342345]\mathbf{A}_{145} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \\ 2 & 3 \\ 4 & 5 \end{smallmatrix}\right].

(9)

A146\mathbf{A}_{146}. A146=[40−5−11001]\mathbf{A}_{146} = \left[\begin{smallmatrix} 4 & 0 \\ -5 & -1 \\ 1 & 0 \\ 0 & 1 \end{smallmatrix}\right].

(10)

A443\mathbf{A}_{443}. A443=[2531174375945313275945413425322048]\mathbf{A}_{443} = \left[\begin{smallmatrix} 25 & 31 & 17 & 43 \\ 75 & 94 & 53 & 132 \\ 75 & 94 & 54 & 134 \\ 25 & 32 & 20 & 48 \end{smallmatrix}\right].

(11)

A587\mathbf{A}_{587}. A587=[1225323−526−43−48−91−41−3−122]\mathbf{A}_{587} = \left[\begin{smallmatrix} 1 & 2 & 2 & 5 & 3 \\ 2 & 3 & -5 & 26 & -4 \\ 3 & -4 & 8 & -9 & 1 \\ -4 & 1 & -3 & -12 & 2 \end{smallmatrix}\right].

(12)

A536\mathbf{A}_{536}. A536=[0701014391010502847503010]\mathbf{A}_{536} = \left[\begin{smallmatrix} 0 & 7 & 0 & 1 & 0 \\ 1 & 4 & 3 & 9 & 1 \\ 0 & 1 & 0 & 5 & 0 \\ 2 & 8 & 4 & 7 & 5 \\ 0 & 3 & 0 & 1 & 0 \end{smallmatrix}\right].

Problem 18.10

Can the given homogeneous system of linear equations and its adjoint system have the same number of linearly independent solutions?

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Problem 18.11

Can the solution sets of the given homogeneous system of linear equations and its adjoint coincide?

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Problem 18.12

Prove that a homogeneous system of linear equations has a nontrivial solution if and only if the rows of the main matrix of the adjoint system are linearly dependent.

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Problem 18.13

Knowing one fundamental matrix Φ\Phi, find the general form of an arbitrary fundamental matrix of the same system of equations.

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Problem 18.14

The given matrix is a fundamental matrix of some homogeneous system of linear equations. Find at least one normal fundamental matrix:

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(1)

A117\mathbf{A}_{117}. A117=[122334]\mathbf{A}_{117} = \left[\begin{smallmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 4 \end{smallmatrix}\right].

(2)

A118\mathbf{A}_{118}. A118=[345723]\mathbf{A}_{118} = \left[\begin{smallmatrix} 3 & 4 \\ 5 & 7 \\ 2 & 3 \end{smallmatrix}\right].

(3)

A397\mathbf{A}_{397}. A397=[013235357340]\mathbf{A}_{397} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \\ 3 & 4 & 0 \end{smallmatrix}\right].

Problem 18.15

The given matrix is a fundamental matrix of some system of linear equations. Find all normal fundamental matrices of this system of equations:

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(1)

A119\mathbf{A}_{119}. A119=[5−4−863−2]\mathbf{A}_{119} = \left[\begin{smallmatrix} 5 & -4 \\ -8 & 6 \\ 3 & -2 \end{smallmatrix}\right].

(2)

c197\mathbf{c}_{197}. c197=[1122]\mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right].

(3)

A112\mathbf{A}_{112}. A112=[001−110]\mathbf{A}_{112} = \left[\begin{smallmatrix} 0 & 0 \\ 1 & -1 \\ 1 & 0 \end{smallmatrix}\right].

(4)

A398\mathbf{A}_{398}. A398=[−1−1−1100010001]\mathbf{A}_{398} = \left[\begin{smallmatrix} -1 & -1 & -1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right].

Problem 18.16

In the system of equations Ax=oA\mathbf{x} = \mathbf{o} (x is a column) having fundamental matrix Φ\Phi, a change of unknowns x=Sy\mathbf{x} = S\mathbf{y} (det⁡S≠0)(\operatorname {det}S \neq 0) is performed. What system of equations results for y? Indicate the fundamental matrix of solutions of this system.

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Problem 18.17

Find at least one homogeneous system of linear equations for which the given matrix is fundamental:

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(1)

A110\mathbf{A}_{110}. A110=[312110]\mathbf{A}_{110} = \left[\begin{smallmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{smallmatrix}\right].

(2)

A147\mathbf{A}_{147}. A147=[101−101−4−1]\mathbf{A}_{147} = \left[\begin{smallmatrix} 1 & 0 \\ 1 & -1 \\ 0 & 1 \\ -4 & -1 \end{smallmatrix}\right].

(3)

c197\mathbf{c}_{197}. c197=[1122]\mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right].

(4)

(р) A148\mathbf{A}_{148}. A148=[11121113]\mathbf{A}_{148} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \\ 1 & 1 \\ 1 & 3 \end{smallmatrix}\right].

(5)

A411\mathbf{A}_{411}. A411=[100213105−1010001]\mathbf{A}_{411} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 2 & 13 & 1 \\ 0 & 5 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right].

Problem 18.18

Find all homogeneous systems of equations equivalent to the given system Ax=oA\mathrm{x} = \mathbf{o}.

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Problem 18.19

Find all homogeneous systems of equations for which the given matrix Φ\Phi is fundamental.

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Problem 18.20

Given a matrix AA whose rows are linearly independent. Below it, the transposed fundamental matrix of the system Ax=A\mathbf{x} = о is appended. Prove that the determinant of the resulting matrix is nonzero.

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