Subsection 19.1

Systems of linear non-homogeneous equations (19.1-19.12)

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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}