Please see image For cach of the following systems, apply Exercise 3 to determine if the systemhas solutions. If there are solutions, find all of them. Finally, find a basis forthe corresponding homogeneous system.X1 + x2 - 3x, + x4 = -2(a)x1 + 2x2 - x3 + x4 = 2(b)2x, + x2 + x, - X4 = 3X1 + 2x2 – 3x, + 2x4 = 22X1 + X2 + x3 – x4 =%3DX1 + X2 - X3%3DX1 + x2 - 3x3 + x4 = 1X1 + x2 + X3 – X4 = 2X1 + X2 - X3(c)

Name: Please see image For cach of the following systems, apply Exercise 3 t
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Description: Please see image For cach of the following systems, apply Exercise 3 to determine if the systemhas solutions. If there are solutions, find all of them. Finally, find a basis forthe corresponding homogeneous system.X1 + x2 - 3x, + x4 = -2(a)x1 + 2x2 -

Answered: For cach of the following systems,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For cach of the following systems, apply Exercise 3 to determine if the system
has solutions. If there are solutions, find all of them. Finally, find a basis for
the corresponding homogeneous system.
X1 + x2 - 3x, + x4 = -2
(a)
x1 + 2x2 - x3 + x4 = 2
(b)
2x, + x2 + x, - X4 = 3
X1 + 2x2 – 3x, + 2x4 = 2
2
X1 + X2 + x3 – x4 =
%3D
X1 + X2 - X3
%3D
X1 + x2 - 3x3 + x4 = 1
X1 + x2 + X3 – X4 = 2
X1 + X2 - X3
(c)

Expert Solution

Step 1

Given :

$x_{1} + 2 x_{2} - x_{3} + x_{4} = 2 2 x_{1} + x_{2} + x_{3} - x_{4} = 3 x_{1} + 2 x_{2} - 3 x_{3} + 2 x_{4} = 2$

To Find : (a) Solution of given system

: (b) Basis for Corresponding Homogeneous system

Note : According to the guideline we are suppose to answer only one at a time . Kindly repost the remaining question and mention it .

Step 2

$(a) [\begin{matrix} 1 & 2 & - 3 & 1 & | & 2 \\ 2 & 1 & 1 & - 1 & | & 3 \\ 1 & 2 & - 3 & 2 & | & 2 \end{matrix}]$

Eliminate the $1^{s t}$ Column

$[\begin{matrix} 1 & 2 & - 3 & 1 & | & 2 \\ 0 & - 3 & 3 & - 3 & | & - 1 \\ 0 & 0 & - 2 & 1 & | & 0 \end{matrix}]$

Make the pivot in the $2^{n d}$ Column by dividing the $2^{n d}$ row by -3 .

$[\begin{matrix} 1 & 2 & - 3 & 1 & | & 2 \\ 0 & 1 & - 1 & 1 & | & \frac{1}{3} \\ 0 & 0 & - 2 & 1 & | & 0 \end{matrix}]$

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