with Unique Solution Budi Murtiyasa Universitas Muhammadiyah Surakarta 1 budi murtiyasa linear equation budi murtiyasa linear equation 2 S y stem of linear Equations 2x 1 ID: 813379
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Slide1
System of Linear Equationswith Unique Solution
Budi MurtiyasaUniversitas Muhammadiyah Surakarta
1
budi murtiyasa / linear equation
Slide2budi murtiyasa / linear equation
2System
of linear Equations
2x
1
– x
2
+ 2x3 = 7 x1 + 3x2 – 5x3 = 0- x1 + x3 = 4
Using matrix
=
A
X
=
G
3x
1
– 7x
2
+ x
3
= 0
-2x
1
+ 3x2 – 4x3 = 0
Using matrix
=
A
X
=
G
A,
coeficient matrix
X,
variable matrix
G,
constant matrix
Slide3budi murtiyasa / linear equation
3SYSTEM OF LINEAR
EQUATIONSA X = G
G
=
0
?
YES
HOMOGENEOUS SYSTEM
A X = 0
NO
NONHOMOGENEOUS SYSTEM
A X
=
G, where G
≠ 0example :
3x – 5y + 3z = 0 x + 2y – z = 0
2x + y + 2z = 0
Example:
2x + y – 7z = 03x + 2y + z = 5 x – 6y + 2z = 0
Slide4budi murtiyasa / linear equation
4Nonhomogeneous S
LE with unique (one) solution
Find the solution of
:
x
1 – 2x2 + x3 = -53x1 + x2 – 2x3 = 11-2x1 + x2 + x
3 = -2
Using a inverse of matrix :
1. Find inverse
of A (by
adjoint matrix).
The solution :
Thus :x
1 = 2x2 = 3x3 = -1
A X = GA
-1 A X = A-1 GX = A-1 G
A
=
,
then
Adj
(A) =
det
(
A) = 6
A-1 =
Adj
(
A)
=
2. X = A
-1
G
X =
=
Slide5budi murtiyasa / linear equation
5Solve the system using inverse matrix:
x
1
– 2x
2
+ x3 = 0 -2x1 + 3x2 – 4x3
= -8
5x1 + x
2 – x3 =
-4
Slide6budi murtiyasa / linear equation
6Using inverse matrices, solve the system:x1 – 2x2
+ x3 = -53x1 + x
2
– 2x
3
= 11
-2x1 + x2 + x3 = -2
Slide7budi murtiyasa / linear equation
7Nonhomogenous SLE with unique
solution
Find a solution of
:
x
1
– 2x2 + x3 = -53x1 + x2 – 2x3 = 11-2x1 + x2 + x
3 = -2
Using Cramer Rule:
finding det
(A), and det
(Ai), that are the determinan
t of
A by replacing the ith
coloumn with constant matrix
G
The solution :
|A| =
= 6
| A1 | =
2. X
i = |A
i | / | A |
= 12
| A
2
| =
= 18
| A
3
| =
= - 6
Slide8budi murtiyasa / linear equation
8Solve the system : using cramer’s
rule -5x
1
+ 4
x
2
– 2x3 = -10 x1
– 2x2 + x
3 = 2 2x1
+ 3x2 – 4x3 =
-8
Slide9budi murtiyasa / linear equation
9
Solve the systems below using:
Inverse matrix
Cramer’s rule
Slide10budi murtiyasa / linear equation
10
Solve the systems below using:
Inverse matrix
Cramer’s rule
Slide11budi murtiyasa / linear equation
11
Solve the systems below using:
Inverse matrix
Cramer’s rule