Section 19

Systems of linear equations of general form

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Subsection 19.1Systems of linear non-homogeneous equations (19.1-19.12)[12/92%]
Problem 19.1

Solve the system of linear equations:

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(1)

2x−3y=42x - 3y = 4

(2)

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

(3)

2x+y+z=42x + y + z = 4, 3x+z=43x + z = 4

(4)

(2+1)x+(2−1)y−2z=1+2(\sqrt{2} + 1) x + (\sqrt{2} - 1) y - \sqrt{2} z = 1 + \sqrt{2}, x+(3−22)y+(2−2)z=1x + (3 - 2\sqrt{2}) y + (\sqrt{2} - 2) z = 1

(5)

x+2y+3z=−4x + 2y + 3z = - 4, 2x+3y+4z=12x + 3y + 4z = 1, 3x+4y+5z=63x + 4y + 5z = 6

(6)

x1+2x2+x3=2x_{1} + 2x_{2} + x_{3} = 2, 2x1+3x2+x4=12x_{1} + 3x_{2} + x_{4} = 1

(7)

5x1+4x2+x3+3x4=−55x_{1} + 4x_{2} + x_{3} + 3x_{4} = - 5, 2x1+x2+x3+4x4=22x_{1} + x_{2} + x_{3} + 4x_{4} = 2, 3x1+2x2+x3+x4=−33x_{1} + 2x_{2} + x_{3} + x_{4} = - 3, x1+3x2−2x3+2x4=−4x_{1} + 3x_{2} - 2x_{3} + 2x_{4} = - 4

(8)

3x1+x2+x3+2x4=−23x_{1} + x_{2} + x_{3} + 2x_{4} = - 2, 5x1+2x3+5x4=−25x_{1} + 2x_{3} + 5x_{4} = - 2, 6x1+x2+5x3+7x4=−46x_{1} + x_{2} + 5x_{3} + 7x_{4} = - 4, 2x1+x2+2x3+2x4=−22x_{1} + x_{2} + 2x_{3} + 2x_{4} = - 2

(9)

x1+x3+x4+x5=6x_{1} + x_{3} + x_{4} + x_{5} = 6, x2+x3+x4+x5=8x_{2} + x_{3} + x_{4} + x_{5} = 8

(10)

6x1+3x2+14x3−2x4+x5=26x_{1} + 3x_{2} + 14x_{3} - 2x_{4} + x_{5} = 2, 20x1+5x2+10x3+4x4+11x5=2020x_{1} + 5x_{2} + 10x_{3} + 4x_{4} + 11x_{5} = 20, 13x1+4x2+12x3+x4+6x5=1113x_{1} + 4x_{2} + 12x_{3} + x_{4} + 6x_{5} = 11, 4x1+7x2+46x3−12x4−7x5=−124x_{1} + 7x_{2} + 46x_{3} - 12x_{4} - 7x_{5} = - 12, x1−2x2−16x3+5x4+4x5=7x_{1} - 2x_{2} - 16x_{3} + 5x_{4} + 4x_{5} = 7.

Problem 19.2

Prove that:

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(1)

the difference of two solutions of an inhomogeneous system of linear equations is a solution of the corresponding homogeneous system

(2)

the sum of any solution of an inhomogeneous system of linear equations and any solution of the corresponding homogeneous system is also a solution of the given inhomogeneous system.

Problem 19.3

By how many units can the rank of the main matrix of the system differ from the rank of the augmented matrix?

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Problem 19.4

Let a system of mm linear equations in nn unknowns be inconsistent, and let its main matrix have rank nn. To what simplest form can this system of equations be reduced by applying the Gauss algorithm to the rows of the augmented matrix?

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Problem 19.5

Formulate a necessary and sufficient condition for a system of mm linear equations in nn unknowns to have a unique solution.

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Problem 19.6

Compose a system of linear equations from the given augmented matrix. Solve the system or establish its inconsistency.

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(1)

∥A232∣c64∥\left\| \mathbf{A}_{232} \mid \mathbf{c}_{64}\right\|. A232=[123234111],c64=[−112]\mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{64} = \left[\begin{smallmatrix} -1 \\ 1 \\ 2 \end{smallmatrix}\right].

(2)

∥A232∣c65∥\left\| \mathbf{A}_{232} \mid \mathbf{c}_{65}\right\|. A232=[123234111],c65=[−415]\mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{65} = \left[\begin{smallmatrix} -4 \\ 1 \\ 5 \end{smallmatrix}\right].

(3)

∥A238∣c66∥\left\| \mathbf{A}_{238} \mid \mathbf{c}_{66}\right\|. A238=[−310−10−74−4−2−33],c66=[111]\mathbf{A}_{238} = \left[\begin{smallmatrix} -3 & 10 & -10 \\ -7 & 4 & -4 \\ -2 & -3 & 3 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(4)

∥A239∣c67∥\left\| \mathbf{A}_{239} \mid \mathbf{c}_{67}\right\|. A239=[012−10−2−220],c67=[012]\mathbf{A}_{239} = \left[\begin{smallmatrix} 0 & 1 & 2 \\ -1 & 0 & -2 \\ -2 & 2 & 0 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right].

(5)

∥A241∣c68∥\left\| \mathbf{A}_{241} \mid \mathbf{c}_{68}\right\|. A241=[6−115−554−99],c68=[−101]\mathbf{A}_{241} = \left[\begin{smallmatrix} 6 & -1 & 1 \\ 5 & -5 & 5 \\ 4 & -9 & 9 \end{smallmatrix}\right], \mathbf{c}_{68} = \left[\begin{smallmatrix} -1 \\ 0 \\ 1 \end{smallmatrix}\right].

(6)

∥A233∣c70∥\left\| \mathbf{A}_{233} \mid \mathbf{c}_{70}\right\|. A233=[1−123−362−24],c70=[396]\mathbf{A}_{233} = \left[\begin{smallmatrix} 1 & -1 & 2 \\ 3 & -3 & 6 \\ 2 & -2 & 4 \end{smallmatrix}\right], \mathbf{c}_{70} = \left[\begin{smallmatrix} 3 \\ 9 \\ 6 \end{smallmatrix}\right].

(7)

∥A206∣c55∥\left\| \mathbf{A}_{206} \mid \mathbf{c}_{55}\right\|. A206=[−111−521176−26−21],c55=[−4109]\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right], \mathbf{c}_{55} = \left[\begin{smallmatrix} -4 \\ 10 \\ 9 \end{smallmatrix}\right].

(8)

∥A206∣c51∥\left\| \mathbf{A}_{206} \mid \mathbf{c}_{51}\right\|. A206=[−111−521176−26−21],c51=[11−1]\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right], \mathbf{c}_{51} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix}\right].

(9)

∥A206T∣c70∥\left\| \mathbf{A}_{206}^{T} \mid \mathbf{c}_{70}\right\|. A206=[−111−521176−26−21],c70=[396]\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right], \mathbf{c}_{70} = \left[\begin{smallmatrix} 3 \\ 9 \\ 6 \end{smallmatrix}\right].

(10)

∥A400∣c162∥\left\| \mathbf{A}_{400} \mid \mathbf{c}_{162}\right\|. A400=[22−111−52−12−122],c162=[9036]\mathbf{A}_{400} = \left[\begin{smallmatrix} 2 & 2 & -1 \\ 1 & 1 & -5 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{smallmatrix}\right], \mathbf{c}_{162} = \left[\begin{smallmatrix} 9 \\ 0 \\ 3 \\ 6 \end{smallmatrix}\right].

(11)

∥A408∣c243∥\left\| \mathbf{A}_{408} \mid \mathbf{c}_{243}\right\|. A408=[242−1−2−115381−2274],c243=[2−1233]\mathbf{A}_{408} = \left[\begin{smallmatrix} 2 & 4 & 2 \\ -1 & -2 & -1 \\ 1 & 5 & 3 \\ 8 & 1 & -2 \\ 2 & 7 & 4 \end{smallmatrix}\right], \mathbf{c}_{243} = \left[\begin{smallmatrix} 2 \\ -1 \\ 2 \\ 3 \\ 3 \end{smallmatrix}\right].

(12)

∥A585T∣c240∥\left\| \mathbf{A}_{585}^{T} \mid \mathbf{c}_{240}\right\|. A585=[1−124−239−142136−911],c240=[2−597−4]\mathbf{A}_{585} = \left[\begin{smallmatrix} 1 & -1 & 2 & 4 & -2 \\ 3 & 9 & -14 & 2 & 1 \\ 3 & 6 & -9 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{240} = \left[\begin{smallmatrix} 2 \\ -5 \\ 9 \\ 7 \\ -4 \end{smallmatrix}\right].

(13)

∥A581T∣c239∥\left\| \mathbf{A}_{581}^{T} \mid \mathbf{c}_{239}\right\|. A581=[135791−23−45211122522],c239=[−12−34−5]\mathbf{A}_{581} = \left[\begin{smallmatrix} 1 & 3 & 5 & 7 & 9 \\ 1 & -2 & 3 & -4 & 5 \\ 2 & 11 & 12 & 25 & 22 \end{smallmatrix}\right], \mathbf{c}_{239} = \left[\begin{smallmatrix} -1 \\ 2 \\ -3 \\ 4 \\ -5 \end{smallmatrix}\right].

(14)

∥A145T∣c14∥\left\| \mathbf{A}_{145}^{T} \mid \mathbf{c}_{14}\right\|. A145=[12342345],c14=[7−1]\mathbf{A}_{145} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \\ 2 & 3 \\ 4 & 5 \end{smallmatrix}\right], \mathbf{c}_{14} = \left[\begin{smallmatrix} 7 \\ -1 \end{smallmatrix}\right].

(15)

∥A506∣c13∥\left\| \mathbf{A}_{506} \mid \mathbf{c}_{13}\right\|. A506=[3−4712−352],c13=[12]\mathbf{A}_{506} = \left[\begin{smallmatrix} 3 & -4 & 7 & 1 \\ 2 & -3 & 5 & 2 \end{smallmatrix}\right], \mathbf{c}_{13} = \left[\begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right].

(16)

∥A149T∣c15∥\left\| \mathbf{A}_{149}^{T} \mid \mathbf{c}_{15}\right\|. A149=[−20100−403],c15=[−23]\mathbf{A}_{149} = \left[\begin{smallmatrix} -2 & 0 \\ 1 & 0 \\ 0 & -4 \\ 0 & 3 \end{smallmatrix}\right], \mathbf{c}_{15} = \left[\begin{smallmatrix} -2 \\ 3 \end{smallmatrix}\right].

(17)

∥A501∣c16∥\left\| \mathbf{A}_{501} \mid \mathbf{c}_{16}\right\|. A501=[187−151−5−611],c16=[17−9]\mathbf{A}_{501} = \left[\begin{smallmatrix} 1 & 8 & 7 & -15 \\ 1 & -5 & -6 & 11 \end{smallmatrix}\right], \mathbf{c}_{16} = \left[\begin{smallmatrix} 17 \\ -9 \end{smallmatrix}\right].

(18)

∥A502∣c17∥\left\| \mathbf{A}_{502} \mid \mathbf{c}_{17}\right\|. A502=[117−31−1−11],c17=[20]\mathbf{A}_{502} = \left[\begin{smallmatrix} 1 & 1 & 7 & -3 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right], \mathbf{c}_{17} = \left[\begin{smallmatrix} 2 \\ 0 \end{smallmatrix}\right].

(19)

∥A510∣c73∥\left\| \mathbf{A}_{510} \mid \mathbf{c}_{73}\right\|. A510=[1−5−61151−43187−15],c73=[−9717]\mathbf{A}_{510} = \left[\begin{smallmatrix} 1 & -5 & -6 & 11 \\ 5 & 1 & -4 & 3 \\ 1 & 8 & 7 & -15 \end{smallmatrix}\right], \mathbf{c}_{73} = \left[\begin{smallmatrix} -9 \\ 7 \\ 17 \end{smallmatrix}\right].

(20)

∥A511∣c74∥\left\| \mathbf{A}_{511} \mid \mathbf{c}_{74}\right\|. A511=[12−7−3−546329524−7−1],c74=[−37141]\mathbf{A}_{511} = \left[\begin{smallmatrix} 1 & 2 & -7 & -3 \\ -5 & 4 & 63 & 29 \\ 5 & 24 & -7 & -1 \end{smallmatrix}\right], \mathbf{c}_{74} = \left[\begin{smallmatrix} -3 \\ 71 \\ 41 \end{smallmatrix}\right].

(21)

∥A512∣c75∥\left\| \mathbf{A}_{512} \mid \mathbf{c}_{75}\right\|. A512=[2−120117−3−111−1],c75=[120]\mathbf{A}_{512} = \left[\begin{smallmatrix} 2 & -1 & 2 & 0 \\ 1 & 1 & 7 & -3 \\ -1 & 1 & 1 & -1 \end{smallmatrix}\right], \mathbf{c}_{75} = \left[\begin{smallmatrix} 1 \\ 2 \\ 0 \end{smallmatrix}\right].

(22)

∥A513∣c62∥\left\| \mathbf{A}_{513} \mid \mathbf{c}_{62}\right\|. A513=[−1−5−4−32−12−15381],c62=[−5−21]\mathbf{A}_{513} = \left[\begin{smallmatrix} -1 & -5 & -4 & -3 \\ 2 & -1 & 2 & -1 \\ 5 & 3 & 8 & 1 \end{smallmatrix}\right], \mathbf{c}_{62} = \left[\begin{smallmatrix} -5 \\ -2 \\ 1 \end{smallmatrix}\right].

(23)

∥A517∣c63∥\left\| \mathbf{A}_{517} \mid \mathbf{c}_{63}\right\|. A517=[31−242−36−58−123],c63=[12−2]\mathbf{A}_{517} = \left[\begin{smallmatrix} 3 & 1 & -2 & 4 \\ 2 & -3 & 6 & -5 \\ 8 & -1 & 2 & 3 \end{smallmatrix}\right], \mathbf{c}_{63} = \left[\begin{smallmatrix} 1 \\ 2 \\ -2 \end{smallmatrix}\right].

(24)

∥A399T∣c50∥\left\| \mathbf{A}_{399}^{T} \mid \mathbf{c}_{50}\right\|. A399=[1222125112−21],c50=[00−3]\mathbf{A}_{399} = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 5 & 1 & 1 \\ 2 & -2 & 1 \end{smallmatrix}\right], \mathbf{c}_{50} = \left[\begin{smallmatrix} 0 \\ 0 \\ -3 \end{smallmatrix}\right].

(25)

∥A444T∣c167∥\left\| \mathbf{A}_{444}^{T} \mid \mathbf{c}_{167}\right\|. A444=[54132114321113−22],c167=[0−112]\mathbf{A}_{444} = \left[\begin{smallmatrix} 5 & 4 & 1 & 3 \\ 2 & 1 & 1 & 4 \\ 3 & 2 & 1 & 1 \\ 1 & 3 & -2 & 2 \end{smallmatrix}\right], \mathbf{c}_{167} = \left[\begin{smallmatrix} 0 \\ -1 \\ 1 \\ 2 \end{smallmatrix}\right].

(26)

∥A520∣c244∥\left\| \mathbf{A}_{520} \mid \mathbf{c}_{244}\right\|. A520=[1002010−300142341154712],c244=[−210−10−1430]\mathbf{A}_{520} = \left[\begin{smallmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 4 \\ 2 & 3 & 4 & 11 \\ 5 & 4 & 7 & 12 \end{smallmatrix}\right], \mathbf{c}_{244} = \left[\begin{smallmatrix} -2 \\ 10 \\ -10 \\ -14 \\ 30 \end{smallmatrix}\right].

(27)

∥A521∣c244∥\left\| \mathbf{A}_{521} \mid \mathbf{c}_{244}\right\|. A521=[1002010−300142341154726],c244=[−210−10−1430]\mathbf{A}_{521} = \left[\begin{smallmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 4 \\ 2 & 3 & 4 & 11 \\ 5 & 4 & 7 & 26 \end{smallmatrix}\right], \mathbf{c}_{244} = \left[\begin{smallmatrix} -2 \\ 10 \\ -10 \\ -14 \\ 30 \end{smallmatrix}\right].

(28)

∥A523∣c241∥\left\| \mathbf{A}_{523} \mid \mathbf{c}_{241}\right\|. A523=[3541493627−1814−156711−4],c241=[25531]\mathbf{A}_{523} = \left[\begin{smallmatrix} 3 & 5 & 4 & 1 \\ 4 & 9 & 3 & 6 \\ 2 & 7 & -1 & 8 \\ 1 & 4 & -1 & 5 \\ 6 & 7 & 11 & -4 \end{smallmatrix}\right], \mathbf{c}_{241} = \left[\begin{smallmatrix} 2 \\ 5 \\ 5 \\ 3 \\ 1 \end{smallmatrix}\right].

(29)

∥A524∣c242∥\left\| \mathbf{A}_{524} \mid \mathbf{c}_{242}\right\|. A524=[2514−5−8−7−1354124221303],c242=[4−1123]\mathbf{A}_{524} = \left[\begin{smallmatrix} 2 & 5 & 1 & 4 \\ -5 & -8 & -7 & -1 \\ 3 & 5 & 4 & 1 \\ 2 & 4 & 2 & 2 \\ 1 & 3 & 0 & 3 \end{smallmatrix}\right], \mathbf{c}_{242} = \left[\begin{smallmatrix} 4 \\ -1 \\ 1 \\ 2 \\ 3 \end{smallmatrix}\right].

(30)

∥A587T∣c245∥\left\| \mathbf{A}_{587}^{T} \mid \mathbf{c}_{245}\right\|. A587=[1225323−526−43−48−91−41−3−122],c245=[−7−57−599]\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], \mathbf{c}_{245} = \left[\begin{smallmatrix} -7 \\ -5 \\ 7 \\ -59 \\ 9 \end{smallmatrix}\right].

(31)

∥A574∣c18∥\left\| \mathbf{A}_{574} \mid \mathbf{c}_{18}\right\|. A574=[2−13−244−2517],c18=[31]\mathbf{A}_{574} = \left[\begin{smallmatrix} 2 & -1 & 3 & -2 & 4 \\ 4 & -2 & 5 & 1 & 7 \end{smallmatrix}\right], \mathbf{c}_{18} = \left[\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}\right].

(32)

∥A575∣c46∥\left\| \mathbf{A}_{575} \mid \mathbf{c}_{46}\right\|. A575=[1−111−221−1−11],c46=[2−4]\mathbf{A}_{575} = \left[\begin{smallmatrix} 1 & -1 & 1 & 1 & -2 \\ 2 & 1 & -1 & -1 & 1 \end{smallmatrix}\right], \mathbf{c}_{46} = \left[\begin{smallmatrix} 2 \\ -4 \end{smallmatrix}\right].

(33)

∥A576∣c33∥\left\| \mathbf{A}_{576} \mid \mathbf{c}_{33}\right\|. A576=[12−1111−213−1],c33=[21]\mathbf{A}_{576} = \left[\begin{smallmatrix} 1 & 2 & -1 & 1 & 1 \\ 1 & -2 & 1 & 3 & -1 \end{smallmatrix}\right], \mathbf{c}_{33} = \left[\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\right].

(34)

∥A581∣c77∥\left\| \mathbf{A}_{581} \mid \mathbf{c}_{77}\right\|. A581=[135791−23−45211122522],c77=[0−11]\mathbf{A}_{581} = \left[\begin{smallmatrix} 1 & 3 & 5 & 7 & 9 \\ 1 & -2 & 3 & -4 & 5 \\ 2 & 11 & 12 & 25 & 22 \end{smallmatrix}\right], \mathbf{c}_{77} = \left[\begin{smallmatrix} 0 \\ -1 \\ 1 \end{smallmatrix}\right].

(35)

∥A581∣c76∥\left\| \mathbf{A}_{581} \mid \mathbf{c}_{76}\right\|. A581=[135791−23−45211122522],c76=[11−3]\mathbf{A}_{581} = \left[\begin{smallmatrix} 1 & 3 & 5 & 7 & 9 \\ 1 & -2 & 3 & -4 & 5 \\ 2 & 11 & 12 & 25 & 22 \end{smallmatrix}\right], \mathbf{c}_{76} = \left[\begin{smallmatrix} 1 \\ 1 \\ -3 \end{smallmatrix}\right].

(36)

∥A422T∣c72∥\left\| \mathbf{A}_{422}^{T} \mid \mathbf{c}_{72}\right\|. A422=[4192−11−15928−20135−342],c72=[−513]\mathbf{A}_{422} = \left[\begin{smallmatrix} 4 & 1 & 9 \\ 2 & -11 & -15 \\ 9 & 2 & 8 \\ -20 & 13 & 5 \\ -3 & 4 & 2 \end{smallmatrix}\right], \mathbf{c}_{72} = \left[\begin{smallmatrix} -5 \\ 1 \\ 3 \end{smallmatrix}\right].

(37)

∥A577∣c78∥\left\| \mathbf{A}_{577} \mid \mathbf{c}_{78}\right\|. A577=[2−13−244−25172−1182],c78=[31−7]\mathbf{A}_{577} = \left[\begin{smallmatrix} 2 & -1 & 3 & -2 & 4 \\ 4 & -2 & 5 & 1 & 7 \\ 2 & -1 & 1 & 8 & 2 \end{smallmatrix}\right], \mathbf{c}_{78} = \left[\begin{smallmatrix} 3 \\ 1 \\ -7 \end{smallmatrix}\right].

(38)

∥A584∣c79∥\left\| \mathbf{A}_{584} \mid \mathbf{c}_{79}\right\|. A584=[21−1−111−111−233−3−34],c79=[20−242]\mathbf{A}_{584} = \left[\begin{smallmatrix} 2 & 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & 1 & -2 \\ 3 & 3 & -3 & -3 & 4 \end{smallmatrix}\right], \mathbf{c}_{79} = \left[\begin{smallmatrix} 20 \\ -2 \\ 42 \end{smallmatrix}\right].

(39)

∥A578∣c80∥\left\| \mathbf{A}_{578} \mid \mathbf{c}_{80}\right\|. A578=[3111−26123−410136−7],c80=[2611]\mathbf{A}_{578} = \left[\begin{smallmatrix} 3 & 1 & 1 & 1 & -2 \\ 6 & 1 & 2 & 3 & -4 \\ 10 & 1 & 3 & 6 & -7 \end{smallmatrix}\right], \mathbf{c}_{80} = \left[\begin{smallmatrix} 2 \\ 6 \\ 11 \end{smallmatrix}\right].

(40)

∥A579∣c81∥\left\| \mathbf{A}_{579} \mid \mathbf{c}_{81}\right\|. A579=[111220111211021],c81=[633]\mathbf{A}_{579} = \left[\begin{smallmatrix} 1 & 1 & 1 & 2 & 2 \\ 0 & 1 & 1 & 1 & 2 \\ 1 & 1 & 0 & 2 & 1 \end{smallmatrix}\right], \mathbf{c}_{81} = \left[\begin{smallmatrix} 6 \\ 3 \\ 3 \end{smallmatrix}\right].

(41)

∥A580∣c82∥\left\| \mathbf{A}_{580} \mid \mathbf{c}_{82}\right\|. A580=[142211−22−2−11202−1],c82=[−57−3]\mathbf{A}_{580} = \left[\begin{smallmatrix} 1 & 4 & 2 & 2 & 1 \\ 1 & -2 & 2 & -2 & -1 \\ 1 & 2 & 0 & 2 & -1 \end{smallmatrix}\right], \mathbf{c}_{82} = \left[\begin{smallmatrix} -5 \\ 7 \\ -3 \end{smallmatrix}\right].

(42)

∥A581∣c58∥\left\| \mathbf{A}_{581} \mid c_{58}\right\|. A581=[135791−23−45211122522]\mathbf{A}_{581} = \left[\begin{smallmatrix} 1 & 3 & 5 & 7 & 9 \\ 1 & -2 & 3 & -4 & 5 \\ 2 & 11 & 12 & 25 & 22 \end{smallmatrix}\right].

(43)

∥A522T∣c163∥\left\| \mathbf{A}_{522}^{T} \mid \mathbf{c}_{163}\right\|. A522=[10−2021−15−110002154−117],c163=[2−211]\mathbf{A}_{522} = \left[\begin{smallmatrix} 1 & 0 & -2 & 0 \\ 2 & 1 & -1 & 5 \\ -1 & 1 & 0 & 0 \\ 0 & 2 & 1 & 5 \\ 4 & -1 & 1 & 7 \end{smallmatrix}\right], \mathbf{c}_{163} = \left[\begin{smallmatrix} 2 \\ -2 \\ 1 \\ 1 \end{smallmatrix}\right].

(44)

∥A523T∣c164∥\left\| \mathbf{A}_{523}^{T} \mid \mathbf{c}_{164}\right\|. A523=[3541493627−1814−156711−4],c164=[1211]\mathbf{A}_{523} = \left[\begin{smallmatrix} 3 & 5 & 4 & 1 \\ 4 & 9 & 3 & 6 \\ 2 & 7 & -1 & 8 \\ 1 & 4 & -1 & 5 \\ 6 & 7 & 11 & -4 \end{smallmatrix}\right], \mathbf{c}_{164} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 1 \end{smallmatrix}\right].

(45)

∥A524T∣c165∥\left\| \mathbf{A}_{524}^{T} \mid \mathbf{c}_{165}\right\|. A524=[2514−5−8−7−1354124221303],c165=[01−12]\mathbf{A}_{524} = \left[\begin{smallmatrix} 2 & 5 & 1 & 4 \\ -5 & -8 & -7 & -1 \\ 3 & 5 & 4 & 1 \\ 2 & 4 & 2 & 2 \\ 1 & 3 & 0 & 3 \end{smallmatrix}\right], \mathbf{c}_{165} = \left[\begin{smallmatrix} 0 \\ 1 \\ -1 \\ 2 \end{smallmatrix}\right].

(46)

∥A588∣c166∥\left\| \mathbf{A}_{588} \mid \mathbf{c}_{166}\right\|. A588=[21−1−111−111−233−3−3445−5−57],c166=[1111]\mathbf{A}_{588} = \left[\begin{smallmatrix} 2 & 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & 1 & -2 \\ 3 & 3 & -3 & -3 & 4 \\ 4 & 5 & -5 & -5 & 7 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(47)

∥A520T∣c170∥\left\| \mathbf{A}_{520}^{T} \mid \mathbf{c}_{170}\right\|. A520=[1002010−300142341154712],c170=[−181714]\mathbf{A}_{520} = \left[\begin{smallmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 4 \\ 2 & 3 & 4 & 11 \\ 5 & 4 & 7 & 12 \end{smallmatrix}\right], \mathbf{c}_{170} = \left[\begin{smallmatrix} -1 \\ 8 \\ 17 \\ 14 \end{smallmatrix}\right].

(48)

∥A544∣c246∥\left\| \mathbf{A}_{544} \mid \mathbf{c}_{246}\right\|. A544=[1230023400111000001200034],c246=[01102]\mathbf{A}_{544} = \left[\begin{smallmatrix} 1 & 2 & 3 & 0 & 0 \\ 2 & 3 & 4 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 3 & 4 \end{smallmatrix}\right], \mathbf{c}_{246} = \left[\begin{smallmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 2 \end{smallmatrix}\right].

(49)

∥A533∣c247∥\left\| \mathbf{A}_{533} \mid \mathbf{c}_{247}\right\|. A533=[1101001031040,1230400500000,1360800000,14900000,15],c247=[300003000100403]\mathbf{A}_{533} = \left[\begin{smallmatrix} 1 & 10 & 100 & 10^{3} & 10^{4} \\ 0,1 & 2 & 30 & 400 & 5000 \\ 0 & 0,1 & 3 & 60 & 800 \\ 0 & 0 & 0,1 & 4 & 90 \\ 0 & 0 & 0 & 0,1 & 5 \end{smallmatrix}\right], \mathbf{c}_{247} = \left[\begin{smallmatrix} 30000 \\ 3000 \\ 100 \\ 40 \\ 3 \end{smallmatrix}\right].

(50)

∥A591∣c271∥\left\| \mathbf{A}_{591} \mid \mathbf{c}_{271}\right\|. A591=[10−1010010−1011−1001−101−1001100−110],c271=[1−12−11]\mathbf{A}_{591} = \left[\begin{smallmatrix} 1 & 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 0 & 1 & -1 \\ 0 & 1 & -1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 & 1 & 0 \end{smallmatrix}\right], \mathbf{c}_{271} = \left[\begin{smallmatrix} 1 \\ -1 \\ 2 \\ -1 \\ 1 \end{smallmatrix}\right].

Problem 19.7

Compose a system of linear equations from the given augmented matrix containing a parameter. Find all values of the parameter for which the system is consistent, and solve it:

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(1)

∥A223∣c85∥\left\| \mathbf{A}_{223} \mid \mathbf{c}_{85}\right\|. A223=[23137−6581],c85=[7−2λ]\mathbf{A}_{223} = \left[\begin{smallmatrix} 2 & 3 & 1 \\ 3 & 7 & -6 \\ 5 & 8 & 1 \end{smallmatrix}\right], \mathbf{c}_{85} = \left[\begin{smallmatrix} 7 \\ -2 \\ \lambda \end{smallmatrix}\right].

(2)

∥A224∣c86∥\left\| \mathbf{A}_{224} \mid \mathbf{c}_{86}\right\|. A224=[−24−3447142],c86=[λ57]\mathbf{A}_{224} = \left[\begin{smallmatrix} -2 & 4 & -3 \\ 4 & 4 & 7 \\ 1 & 4 & 2 \end{smallmatrix}\right], \mathbf{c}_{86} = \left[\begin{smallmatrix} \lambda \\ 5 \\ 7 \end{smallmatrix}\right].

(3)

∥A225∣c87∥\left\| \mathbf{A}_{225} \mid \mathbf{c}_{87}\right\|. A225=[2−111−43−1−32],c87=[2λ5]\mathbf{A}_{225} = \left[\begin{smallmatrix} 2 & -1 & 1 \\ 1 & -4 & 3 \\ -1 & -3 & 2 \end{smallmatrix}\right], \mathbf{c}_{87} = \left[\begin{smallmatrix} 2 \\ \lambda \\ 5 \end{smallmatrix}\right].

(4)

∥A226∣c88∥\left\| \mathbf{A}_{226} \mid \mathbf{c}_{88}\right\|. A226=[322244577],c88=[56λ]\mathbf{A}_{226} = \left[\begin{smallmatrix} 3 & 2 & 2 \\ 2 & 4 & 4 \\ 5 & 7 & 7 \end{smallmatrix}\right], \mathbf{c}_{88} = \left[\begin{smallmatrix} 5 \\ 6 \\ \lambda \end{smallmatrix}\right].

Problem 19.8

Describe all linear combinations of solutions of the given inhomogeneous system of linear equations that are solutions of the same system.

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Problem 19.9

Describe all such linear combinations of solutions of the given linear inhomogeneous system of equations that are solutions of the corresponding homogeneous system.

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Problem 19.10

Let the column of constant terms of a linear system of equations equal the sum of the columns of its main matrix. Indicate some particular solution of the system.

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Problem 19.11

Let the column of constant terms of a linear system of equations coincide with the last column of its main matrix. Indicate some particular solution of the system.

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Problem 19.12

Let x,y\mathbf{x}, \mathbf{y} be columns of solutions of the systems of equations Ax=a,Ay=bA\mathbf{x} = \mathbf{a}, A\mathbf{y} = \mathbf{b} respectively, and let α,β\alpha , \beta be some numbers. What system of equations does the following satisfy:

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(1)

z=αx\mathbf{z} = \alpha \mathbf{x}

(2)

z=x+y\mathbf{z} = \mathbf{x} + \mathbf{y}

(3)

z=αx+βy\mathbf{z} = \alpha \mathbf{x} + \beta \mathbf{y}

Subsection 19.2Conditions for the consistency of the system of linear equations (19.13-19.20)[8/63%]
Problem 19.13

Prove that if the columns of the main matrix are linearly independent, then the system of linear equations has at most one solution.

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Problem 19.14

Prove that if the rows of the main matrix are linearly independent, then the system of equations is consistent for any column of constant terms.

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Problem 19.15

Prove the following statement: if a system of linear equations is consistent for any column of constant terms, then the rows of its main matrix are linearly independent.

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Problem 19.16

Prove that exactly one of two possibilities holds: either the system of linear equations Ax=bA\mathbf{x} = \mathbf{b} is consistent for any column of constant terms, or its adjoint homogeneous system ATy=0A^{T} \mathbf{y} = 0 has a nonzero solution (the Fredholm alternative).

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Problem 19.17

Formulate conditions (and prove their necessity and sufficiency) that the main matrix must satisfy so that the number of solutions of the system of linear equations, depending on the column bb of constant terms, equals:

?
(1)

0 or 1

(2)

1 or ∞\infty

(3)

0 or ∞\infty

(4)

1 for all b.

Problem 19.18

A system of linear equations is given by its augmented matrix. Check the consistency of this system, using the Fredholm theorem and the result of problem 18.9 for the corresponding adjoint system of equations:

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(1)

∥A114∣c89∥\left\| \mathbf{A}_{114} \mid \mathbf{c}_{89}\right\|. A114=[355947],c89=[12−1]\mathbf{A}_{114} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \\ 4 & 7 \end{smallmatrix}\right], \mathbf{c}_{89} = \left[\begin{smallmatrix} 1 \\ 2 \\ -1 \end{smallmatrix}\right].

(2)

∥A118∣c90∥\left\| \mathbf{A}_{118} \mid \mathbf{c}_{90}\right\|. A118=[345723],c90=[231]\mathbf{A}_{118} = \left[\begin{smallmatrix} 3 & 4 \\ 5 & 7 \\ 2 & 3 \end{smallmatrix}\right], \mathbf{c}_{90} = \left[\begin{smallmatrix} 2 \\ 3 \\ 1 \end{smallmatrix}\right].

(3)

∥A587∣c171∥\left\| \mathbf{A}_{587} \mid \mathbf{c}_{171}\right\|. A587=[1225323−526−43−48−91−41−3−122],c171=[−1110]\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], \mathbf{c}_{171} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 0 \end{smallmatrix}\right].

Problem 19.19

A system of equations is given by its augmented matrix containing a parameter. Using the Fredholm theorem, find all values of the parameter for which the system is consistent, and solve it:

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(1)

∥A205∣c91∥\left\| \mathbf{A}_{205} \mid \mathbf{c}_{91}\right\|. A205=[123234345],c91=[αα2α3]\mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right], \mathbf{c}_{91} = \left[\begin{smallmatrix} \alpha \\ \alpha^{2} \\ \alpha^{3} \end{smallmatrix}\right].

(2)

∥A515∣c91∥\left\| \mathbf{A}_{515} \mid \mathbf{c}_{91}\right\|. A515=[3−2546−4439−632],c91=[αα2α3]\mathbf{A}_{515} = \left[\begin{smallmatrix} 3 & -2 & 5 & 4 \\ 6 & -4 & 4 & 3 \\ 9 & -6 & 3 & 2 \end{smallmatrix}\right], \mathbf{c}_{91} = \left[\begin{smallmatrix} \alpha \\ \alpha^{2} \\ \alpha^{3} \end{smallmatrix}\right].

(3)

∥A507∣c91∥\left\| \mathbf{A}_{507} \mid \mathbf{c}_{91}\right\|. A507=[134122314011],c91=[αα2α3]\mathbf{A}_{507} = \left[\begin{smallmatrix} 1 & 3 & 4 & 1 \\ 2 & 2 & 3 & 1 \\ 4 & 0 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{91} = \left[\begin{smallmatrix} \alpha \\ \alpha^{2} \\ \alpha^{3} \end{smallmatrix}\right].

Problem 19.20

A system of equations is given by its augmented matrix ∥A644∣c282∥\left\| \mathbf{A}_{644} \mid \mathbf{c}_{282}\right\|, depending on the parameters λ1,…,λn,μ\lambda_{1}, \ldots , \lambda_{n}, \mu. Describe the set of values of the parameters for which the system is consistent, and solve it. A644=[111…1λ1λ2λ3…λnλ12λ22λ32…λn2…………………λ1n−1λ2n−1λ3n−1…λnn−1],c282=[1μμ2⋮μn−1]\mathbf{A}_{644} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 \\ \lambda_1 & \lambda_2 & \lambda_3 & \ldots & \lambda_n \\ \lambda_1^2 & \lambda_2^2 & \lambda_3^2 & \ldots & \lambda_n^2 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \ldots \\ \lambda_1^{n-1} & \lambda_2^{n-1} & \lambda_3^{n-1} & \ldots & \lambda_n^{n-1} \end{smallmatrix}\right], \mathbf{c}_{282} = \left[\begin{smallmatrix} 1 \\ \mu \\ \mu^2 \\ \vdots \\ \mu^{n-1} \end{smallmatrix}\right].

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Subsection 19.3Equivalent systems of equations (19.21-19.29)[9/56%]
Problem 19.21

Prove that if consistent systems of linear inhomogeneous equations are equivalent, then the corresponding homogeneous systems are also equivalent.

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Problem 19.22
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(1)

Prove that the nontrivial equations1 a1x1+…+anxn=0a_{1} x_{1} + \ldots + a_{n} x_{n} = 0 and b1x1+⋯+bnxn=0b_{1} x_{1} + \cdots + b_{n} x_{n} = 0 are equivalent if and only if rg⁡[a1…anb1…bn]=1\operatorname {rg}\left[\begin{smallmatrix} a_{1} & \ldots & a_{n} \\ b_{1} & \ldots & b_{n}\end{smallmatrix}\right] = 1.

Footnotes

  1. A linear equation is nontrivial if at least one of the coefficients of the unknowns is different from 0. ↩

(2)

Prove that the nontrivial1{ }^{1}) equations a1x1+…⋯+anxn=aa_{1} x_{1} + \ldots \cdots + a_{n} x_{n} = a and b1x1+⋯+bnxn=bb_{1} x_{1} + \cdots + b_{n} x_{n} = b are equivalent if and only if rg⁡[a1…anab1…bnb]=1\operatorname {rg}\left[\begin{smallmatrix} a_{1} & \ldots & a_{n} & a \\ b_{1} & \ldots & b_{n} & b\end{smallmatrix}\right] = 1.

(3)

Formulate a criterion for pairwise equivalence of kk linear equations.

Problem 19.23
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(1)

Prove that the systems of linear equations Ax=o,Bx=oA\mathbf{x} = \mathbf{o}, B\mathbf{x} = \mathbf{o} are equivalent if and only if

rg⁡[AB]□=rg⁡A=rg⁡B. \operatorname {rg}\left[\begin{smallmatrix} A \\ B\end{smallmatrix}\right]^{\square } = \operatorname {rg} A = \operatorname {rg} B.
(2)

Prove that the consistent systems of linear equations Ax=a,Bx=bA\mathbf{x} = \mathbf{a}, B\mathbf{x} = \mathbf{b} are equivalent if and only if

rg⁡[AaB b]□=rg⁡A=rg⁡B. \operatorname {rg}\left[\begin{smallmatrix} A & \mathbf{a} \\ B & \mathrm{~ b} \end{smallmatrix}\right]^{\square } = \operatorname {rg} A = \operatorname {rg} B.
Problem 19.24

Check the equivalence of the systems of equations 18.1,1118. 1, 11) and 18.1,12)18. 1, 12).

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Problem 19.25

Check whether the systems of equations defined by the augmented matrices are equivalent:

?
(1)

(A502∣c17)\left(\mathbf{A}_{502} \mid \mathbf{c}_{17}\right) and (A503∣c12)\left(\mathbf{A}_{503} \mid \mathbf{c}_{12}\right). A502=[117−31−1−11],c17=[20],A503=[3−2112−120],c12=[11]\mathbf{A}_{502} = \left[\begin{smallmatrix} 1 & 1 & 7 & -3 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right], \mathbf{c}_{17} = \left[\begin{smallmatrix} 2 \\ 0 \end{smallmatrix}\right], \mathbf{A}_{503} = \left[\begin{smallmatrix} 3 & -2 & 1 & 1 \\ 2 & -1 & 2 & 0 \end{smallmatrix}\right], \mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right].

(2)

(A239∣c67)\left(\mathbf{A}_{239} \mid \mathbf{c}_{67}\right) and (A240∣c147)\left(\mathbf{A}_{240} \mid \mathbf{c}_{147}\right). A239=[012−10−2−220],c67=[012],A240=[−54−22161−2−2],c147=[5−2−1]\mathbf{A}_{239} = \left[\begin{smallmatrix} 0 & 1 & 2 \\ -1 & 0 & -2 \\ -2 & 2 & 0 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{A}_{240} = \left[\begin{smallmatrix} -5 & 4 & -2 \\ 2 & 1 & 6 \\ 1 & -2 & -2 \end{smallmatrix}\right], \mathbf{c}_{147} = \left[\begin{smallmatrix} 5 \\ -2 \\ -1 \end{smallmatrix}\right].

(3)

(A581∣c69)\left(\mathbf{A}_{581} \mid \mathbf{c}_{69}\right) and (A422T∣c72)\left(\mathbf{A}_{422}^{T} \mid \mathbf{c}_{72}\right). A581=[135791−23−45211122522],c69=[−26−11−67],A422=[4192−11−15928−20135−342],c72=[−513]\mathbf{A}_{581} = \left[\begin{smallmatrix} 1 & 3 & 5 & 7 & 9 \\ 1 & -2 & 3 & -4 & 5 \\ 2 & 11 & 12 & 25 & 22 \end{smallmatrix}\right], \mathbf{c}_{69} = \left[\begin{smallmatrix} -26 \\ -11 \\ -67 \end{smallmatrix}\right], \mathbf{A}_{422} = \left[\begin{smallmatrix} 4 & 1 & 9 \\ 2 & -11 & -15 \\ 9 & 2 & 8 \\ -20 & 13 & 5 \\ -3 & 4 & 2 \end{smallmatrix}\right], \mathbf{c}_{72} = \left[\begin{smallmatrix} -5 \\ 1 \\ 3 \end{smallmatrix}\right].

Problem 19.26

Check the equivalence of all systems in the given collection (each system of equations is given by an augmented matrix): (A501∣c16),(A509∣c67)\left(\mathbf{A}_{501} \mid \mathbf{c}_{16}\right), \left(\mathbf{A}_{509} \mid \mathbf{c}_{67}\right) and (A510∣c73)\left(\mathbf{A}_{510} \mid \mathbf{c}_{73}\right). A501=[187−151−5−611],c16=[17−9],A509=[2−1−34−325−7011−2],c67=[012],A510=[1−5−61151−43187−15],c73=[−9717]\mathbf{A}_{501} = \left[\begin{smallmatrix} 1 & 8 & 7 & -15 \\ 1 & -5 & -6 & 11 \end{smallmatrix}\right], \mathbf{c}_{16} = \left[\begin{smallmatrix} 17 \\ -9 \end{smallmatrix}\right], \mathbf{A}_{509} = \left[\begin{smallmatrix} 2 & -1 & -3 & 4 \\ -3 & 2 & 5 & -7 \\ 0 & 1 & 1 & -2 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{A}_{510} = \left[\begin{smallmatrix} 1 & -5 & -6 & 11 \\ 5 & 1 & -4 & 3 \\ 1 & 8 & 7 & -15 \end{smallmatrix}\right], \mathbf{c}_{73} = \left[\begin{smallmatrix} -9 \\ 7 \\ 17 \end{smallmatrix}\right].

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Problem 19.27
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(1)

Suppose that adjoining to a given homogeneous system of linear equations some further number of linear homogeneous equations does not change the set of its solutions. Prove that the adjoined equations are linear combinations of the equations of the given system.

(2)

Prove the same statement for a consistent system of linear inhomogeneous equations. Compare with problem 17. 3 2)

Problem 19.28
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(1)

Suppose that every solution of the homogeneous system of linear equations (A) is also a solution of the homogeneous system of linear equations (B). Prove that then every equation of system (B) is a linear combination of the equations of system (A).

(2)

Prove the same statement for consistent systems of linear inhomogeneous equations. Compare with problem 17.3(1).

Problem 19.29
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(1)

Prove that two homogeneous systems of linear equations are equivalent if and only if the equations of each of them are linear combinations of the equations of the other system.

(2)

Prove the same statement for consistent systems of linear inhomogeneous equations.

Subsection 19.4Applications (19.30-19.33)[4/50%]
Problem 19.30

Let Ax=bA\mathrm{x} = \mathrm{b} be an arbitrary system of linear equations. Prove that the system of equations (ATA)x=ATb\left(A^{T} A\right) \mathbf{x} = A^{T} \mathbf{b} is consistent.

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Problem 19.31

Given a square matrix A=∥aik∥A = \left\| a_{ik}\right\|. Prove that if for all ii the inequality ∣aii∣>∑k≠i∣aik∣\left|a_{ii}\right| > \sum_{k\neq i}\left|a_{ik}\right| holds, then det⁡A≠0\operatorname {det} A \neq 0.

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Problem 19.32

Prove that for any pairwise distinct numbers a1,…,an+1a_{1}, \ldots , a_{n + 1} and any numbers b1,…,bn+1b_{1}, \ldots , b_{n + 1} there exists a unique polynomial f(t)f(t) of degree at most nn such that

f(a1)=b1,…,f(an+1)=bn+1. f\left(a_{1}\right) = b_{1}, \ldots , f\left(a_{n + 1}\right) = b_{n + 1}.
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Problem 19.33

Find a polynomial f(t)f(t) of the third degree such that f(1)=f(2)=f(3)=1,f(4)=7f(1) = f(2) = f(3) = 1, f(4) = 7.

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Subsection 19.5Applications (19.34-19.49). The problems below, numbered 19.34-19.49, relate to a line, a circle, a plane, and a sphere. One should take the algebraic equation of the corresponding set as the definition and apply the theory of systems of linear equations[16/100%]
Problem 19.34
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(1)

(р) Formulate in terms of ranks and prove a condition on the Cartesian coordinates (a1,b1),(a2,b2),(a3,b3)\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right), \left(a_{3}, b_{3}\right) of three points of the plane, necessary and sufficient for these points not to lie on one line.

(2)

Solve the same question for four points of the plane.

Problem 19.35

Prove that through two distinct points with Cartesian coordinates (a1,b1),(a2,b2)\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right) there passes a unique line, and find its equation.

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Problem 19.36

Show that through three points with coordinates (a1,b1),(a2,b2),(a3,b3)\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right), \left(a_{3}, b_{3}\right), not lying on one line, there passes a unique circle, and find its equation. The coordinate system is Cartesian rectangular.

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Problem 19.37
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(1)

Three lines are given in the plane in a general Cartesian coordinate system by the equations Aix+Biy+Ci=0A_{i} x + B_{i} y + C_{i} = 0, i=1,2,3i = 1, 2, 3. Formulate in terms of ranks and prove a condition on the coefficients of the equations, necessary and sufficient for these lines not to pass through one point.

(2)

Solve the same question for four lines.

Problem 19.38

Using the result of problem 19. 37, determine whether the given lines have a common point:

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(1)

2x+3y+1=0,7x+11y+4=0,3x+4y+1=02x + 3y + 1 = 0, 7x + 11y + 4 = 0, 3x + 4y + 1 = 0

(2)

x+8y+1=0,7x−y+1=0,11x−26y−1=0x + 8y + 1 = 0, 7x - y + 1 = 0, 11x - 26y - 1 = 0, 8x+7y+2=08x + 7y + 2 = 0.

Problem 19.39
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(1)

Four points are given by their Cartesian coordinates (ai,bi,ci),i=1,2,3,4\left(a_{i}, b_{i}, c_{i}\right), i = 1, 2, 3, 4. Formulate in terms of ranks and prove a condition, necessary and sufficient for these points not to lie in one plane.

(2)

Solve the same question for five points.

Problem 19.40

Using the result of problem 19. 39, determine whether the given points lie in one plane:

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(1)

(7,−1,2),(2,3,1),(0,10,0),(3,4,1),(6,−2,2)(7, - 1, 2), (2, 3, 1), (0, 10, 0), (3, 4, 1), (6, - 2, 2)

(2)

(6,1,2),(2,3,1),(3,4,1),(6,2,2)(6, 1, 2), (2, 3, 1), (3, 4, 1), (6, 2, 2).

Problem 19.41

Show that through four points with coordinates (ai,bi,ci),i=1,2,3,4\left(a_{i}, b_{i}, c_{i}\right), \quad i = 1, 2, 3, 4, not lying in one plane, there passes a unique sphere, and find its equation. The coordinate system is Cartesian rectangular.

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Problem 19.42

Three points are given by their Cartesian coordinates (ai,bi,ci),i=1,2,3\left(a_{i}, b_{i}, c_{i}\right), i = 1, 2, 3.

?
(1)

(р) Formulate in terms of ranks and prove a condition, necessary and sufficient for these points not to lie on one line.

(2)

Solve the same question for mm points (m⩾4)(m \geqslant 4).

Problem 19.43

Using the result of problem 19. 42, determine whether the given points lie on one line:

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(1)

(2,3,1),(3,4,2),(0,1,−1),(−2,−1,−3),(−6,−5,−7)(2, 3, 1), (3, 4, 2), (0, 1, - 1), (- 2, - 1, - 3), (- 6, - 5, - 7)

(2)

(2,3,1),(3,4,2),(0,1,1),(2,1,3),(6,5,4)(2, 3, 1), (3, 4, 2), (0, 1, 1), (2, 1, 3), (6, 5, 4).

Problem 19.44

Prove that through three points with Cartesian coordinates (ai,bi,ci),i=1,2,3\left(a_{i}, b_{i}, c_{i}\right), i = 1, 2, 3, not lying on one line, there passes a unique plane, and find its equation.

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Problem 19.45

Two planes are given in a general Cartesian coordinate system by the equations Aix+Biy+Ciz+Di=0,i=1,2A_{i} x + B_{i} y + C_{i} z + D_{i} = 0, i = 1, 2. Formulate in terms of ranks and prove the conditions on the coefficients of the equations, necessary and sufficient for these planes to:

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(1)

coincide

(2)

have a unique common line

(3)

be parallel, but not coincide.

Problem 19.46

Three planes are given in a general Cartesian coordinate system by the equations Aix+Biy+Ciz+Di=0,i=1,2,3A_{i} x + B_{i} y + C_{i} z + D_{i} = 0, i = 1, 2, 3. Formulate in terms of ranks and prove the conditions on the coefficients of the equations, necessary and sufficient for these planes to:

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(1)

coincide

(2)

have a unique common point

(3)

have a unique common line

(4)

be parallel, but not all coincide

(5)

form a prism.

Problem 19.47

Using the result of problem 19. 46, determine the mutual arrangement of the planes:

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(1)

3x+2y+5z−1=0,2x+3y+3z+1=03x + 2y + 5z - 1 = 0, \quad 2x + 3y + 3z + 1 = 0, 9x+16y+13z+1=09x + 16y + 13z + 1 = 0

(2)

x−y−z+1=0,5x−21y−17z+1=0x - y - z + 1 = 0, \quad 5x - 21y - 17z + 1 = 0, 6x−26y−21z+1=06x - 26y - 21z + 1 = 0.

Problem 19.48

Four planes are given in a general Cartesian coordinate system by the equations Aix+Biy+Ciz+Di=0,i=1,2A_{i} x + B_{i} y + C_{i} z + D_{i} = 0, i = 1, 2, 3, 4. It is known that the pairs corresponding to i=1,2i = 1, 2 and i=3,4i = 3, 4 define straight lines. Formulate in terms of ranks and prove the conditions on the coefficients of the equations, necessary and sufficient for these lines to:

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(1)

intersect

(2)

be parallel, but not coincide

(3)

coincide

(4)

be skew.

Problem 19.49

Using the result of problem 19. 48, determine the mutual arrangement of the lines:

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(1)

{3x+2y+5z−1=0−x+2y+3z−1=0\begin{cases} 3x + 2y + 5z - 1 = 0 \\ - x + 2y + 3z - 1 = 0 \end{cases} {4x−6y+7z+2=0,5x+3y−8z−3=0\begin{cases} 4x - 6y + 7z + 2 = 0, \\ 5x + 3y - 8z - 3 = 0 \end{cases}

(2)

{x+3y+3z−2=0x−9y−6z−5=0\begin{cases} x + 3y + 3z - 2 = 0 \\ x - 9y - 6z - 5 = 0 \end{cases} {2x−14y−9z−9=04x+2y+z−7=0.\begin{cases} 2x - 14y - 9z - 9 = 0 \\ 4x + 2y + z - 7 = 0. \end{cases}