Solution of Linear Equations

System of Linear Equation
Definition 1
Let the system of linear equation as
a11x1, a12x2… ,a1nxn = b1
a21x1, a22x2… ,a2nxn = b2
…………………….. ………………………….. (8)
…………………….. Am1x1, am2x2… ,amnxn = bm
Can put the apove system in matrix form as:-

? a11

a12

? ? ?

a1n ?? x1 ?

? b1 ?

? ?? ? ? ?

? a21

a22

? ? ?

a2n ?? x2 ?

? b2 ?

? ? ?

? ? ?

? ?? ? ? ? ?

? ? ………………………….. (8)

? ?? ? ? ?

? ? ?

? ? ?

? ?? ? ?

? ? ?

? ?? ? ? ?

? am1
Or

am1

? ? ?

amn ?? xn ?

? bm ?

AX=B, …………………………………….…………………….. (8)

? a11

a12

? ? ?

a1n ?

? b1 ?

? x1 ?

? ? ? ? ? ?

? a 21

a 22

? ? ?

a2n ?

? b2 ?

? x 2 ?

? ? ? ? ? ?

A= ? ?

? ? ? ?

? ? , B= ?

? ? , and X= ? ? ?

? ? ?
?

? ? ? ? ?
?

? ? ?
? ?

? ? ?
? ?

? am1

am1

? ? ?

amn ?

? bm ?

? xn ?

Where A=mxn, matrix, a11, a12…, amn are constant, X= nx1, B = mx1and,
b1, b2…, bm, are constant x1, x2… xn, variable.
Now we study the following methods {Cramer s Rule, Inverse
Matrices, and Elimination Method}
10-Cramer s Rule

To solve the system (8) by Cramer s Rule. Find determinate of A (| A|)
such that | A| ? 0. Let




| A| =D=

a11
a 21
?
?
a m1

a12
a 22
?
?
am1

? ? ?
? ? ?
? ? ?
? ? ?
? ? ?

a1n
a 2n
?
?
a mn

b1
b2
, D1 = ?
?
bm

a12
a 22
?
?
a m1

? ? ?
? ? ?
? ? ?
? ? ?
? ? ?

a1n
a2n
? ,
?
amn








D2 =

a11 b1
a 21 b2
? ?
? ?
a m1 bm

? ? ?
? ? ?
? ? ?
? ? ?
? ? ?

a1n
a2n
?
?
amn




,..., Dn =

a11
a 21
?
?
a m1

a12
a 22
?
?
am1

? ? ? b1
? ? ? b2
? ? ? ? ,
? ? ? ?
? ? ? bm

To solve system (8), we must find unknown x1, x2… xn as

D
x1 =
D

D
, x2 =
D

D
, … xn = .
D


11-Solution of Linear Equations by using Inverse Matrices To solve the system (8) by using Inverse Matrices Find determinate of A (| A|) such that | A| ? 0.
Or
AX=B,
Turing to the relation between the solution of linear equation and matrix inversion multiplying both sides by A-1 thus
A-1 [AX=B]
A-1 AX= A-1 B.
X= A-1 B.
This equation gives the values of the entire unknown X by a simple multiplication of matrix A by inverse of it matrix. As see in the following example
Example12
Use the matrix inversion method; find the values of (x1, x2, x3) for the following set of linear algebraic equations:-
3x1 -6x2 + 7x3 = 3
4x1 -5x3 = 3……………………………. (9)
5x1 - 8x2 +6x3 = -4
Solution
Put the system (9) in the following matrix form as
AX=B,

? 3 ? 6

7 ?? x1 ?

? 3 ?

? ?? ? ? ?

? 4 0

? 5 ?? x 2 ? ? ? 3 ?

? ?? ? ? ?

? 5 ? 8

6 ?? x 3 ?

? ? 4 ?

Where| A|

3
| A| = 4
5

? 6 7
0 ? 5 ? 462? 0.
? 8 6

We can find the inverse matrix of A (A-1), by any method.



? 0.26
? A-1 = ? 0.52
? 0.48
A-1 [AX=B]

0.14
0.12
0.04

? 0.2 ?
?
? 0.52 ? , now we can see the following
? 0.36 ?

A-1 AX= A-1 B.
X= A-1 B.

? x1 ?

? 0.26

0.14

? 0.2 ?? 3 ?

? ? ?

?? ?

? X= ? x2 ? ? ? 0.52

0.12

? 0.52 ?? 3 ?

? ? ?

?? ?

? x3 ?

? 0.48

0.04

? 0.36 ?? ? 4 ?




? x1 ?

? 2 ?

? ? ? ?
? X= ? x 2 ? ? ? 4 ? , which gives the solution of system as x1 = 2,
? ? ? ?

? x 3 ?

? 3 ?

x2 = 4, x3 = -4.

12-Gauss Elimination Method
We can use Gauss Elimination Method to solve the system of linear
equation in (8), as see in the following example
Example 13
3x1 -x2 + 2x3 = 12
3x1 +2x2 +3x3 = 11……………………………. (10)
2x1 - 2x2 - x3 = 2
Solution
Put the system (10) in the following matrix form


3 -1 2 : 12 R1
3 2 3 : 11 R2 ................................................................. (11)
2 -2 -1 : 12 R3

Where Ri (i= 1, 2, 3) row of system.
Step 1

By using
R2 – R1, and 3R3 -2R1
System (11) become


3 -1 2 : 12 R1
0 7 7 : 21 R2 ............................................................ (12)
0 -4 -7 : -8 R3




Step 2
By using
7R3 +4R2
System (11) become


3 -1 2 : 12 R1
0 7 7 : 21 R2 ............................................................ (13)
0 0 -21 : -42 R3

Step 3
From last system (13) we the following equation
3x1 -x2 -2x3 = 12
7x2 +7x3 = 21
-21x3 = -42
Which can easily to solve this system to find:-
x3 = 2, x2 = 1, x1 = 3.