x+y+z+w=13 2x + 3y -w=-1 -3x + 4y +z+ 2w = 10 x +2y-z+ w = 1
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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Question
HANDWRITTEN THEN BOX THE FINAL ANSWERS (PLS STRICTLY FOLLOW HOW IT WAS SOLVED IN THE SECOND PIC)
given 1: i need gaussian elimination and gauss-jordan reduction
Transcribed Image Text:(3)
x+y+z+w=13
2x + 3y
-W=-1
-3x + 4y +z+ 2w = 10
x +2y-z + w = 1
answer should be
x=2
y=0
Z=6
w=5
![[5 0 -21
5
2
0
L1
5
0
=
5 R3 = 2R₁ + R3
4
0
5
L2(5) + 1 2(0)+2 2(-2) + 4]
5 0
0 5 5
L11 4 0
This is usually done to make the value of a matrix element be equal to zero.
-2
5
Methods of Solving Systems of Linear Equations using Matrices
Gaussian Elimination
A matrix is said to be in row echelon form if:
1) The leading element or the first non-zero element of each row is equal to 1.
2) The leading element of each row is always at the right of the leading element of the previous row.
3) All rows consisting of only zeroes (if any) are at the bottom of the matrix.
x₁ + a₁2x₂+...+ a₁nxn = a₁n+1
x₂ + + a₂nxn=a2n+1
⠀
⠀
Xn = ann+1
1
-3-2
2 -4 -3
-3
6 8
1
[1
0 1
⠀
0 0
a12
6
8
...
To use Gaussian Elimination, we must use elementary row operations to transform the given augmented
matrix of the system of linear equations to its row echelon form.
⠀ S
Examples:
Solve for the values given the systems of linear equations using Gaussian Elimination.
1) x-3y2z = 6
2x - 4y - 3z = 8
-3x + 6y + 8z = -5
ain ain+1
azn azn+1
⠀
!
1
ann+1J
First, let us get the coefficient matrix, variable vector, and constant vector:
1 -3 -21
6
2 -4 -3
-3 6 8
[]-[]
= 8
with the augmented matrix,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94519a40-e312-41b7-b285-e2de4f9c0715%2Fd52ebc4b-0b90-4364-81f6-251eee0124a0%2F8apqk8t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[5 0 -21
5
2
0
L1
5
0
=
5 R3 = 2R₁ + R3
4
0
5
L2(5) + 1 2(0)+2 2(-2) + 4]
5 0
0 5 5
L11 4 0
This is usually done to make the value of a matrix element be equal to zero.
-2
5
Methods of Solving Systems of Linear Equations using Matrices
Gaussian Elimination
A matrix is said to be in row echelon form if:
1) The leading element or the first non-zero element of each row is equal to 1.
2) The leading element of each row is always at the right of the leading element of the previous row.
3) All rows consisting of only zeroes (if any) are at the bottom of the matrix.
x₁ + a₁2x₂+...+ a₁nxn = a₁n+1
x₂ + + a₂nxn=a2n+1
⠀
⠀
Xn = ann+1
1
-3-2
2 -4 -3
-3
6 8
1
[1
0 1
⠀
0 0
a12
6
8
...
To use Gaussian Elimination, we must use elementary row operations to transform the given augmented
matrix of the system of linear equations to its row echelon form.
⠀ S
Examples:
Solve for the values given the systems of linear equations using Gaussian Elimination.
1) x-3y2z = 6
2x - 4y - 3z = 8
-3x + 6y + 8z = -5
ain ain+1
azn azn+1
⠀
!
1
ann+1J
First, let us get the coefficient matrix, variable vector, and constant vector:
1 -3 -21
6
2 -4 -3
-3 6 8
[]-[]
= 8
with the augmented matrix,
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