Linear Equations and Matrix Operations
Example :
Find x
1
and x
2
from these equation :
Solution :
General form :
Where :
a
1
,
a
2
,
a
3
..
a
n
and
b
are constantas
x
1
, x
2
, x
3
.. x
n
are variables
Linear Equation in matrix form
If we have some equations :
Then, we can write :
General form :
Where :
Or, we can write :
Example :
Find x
1
and x
2
from these equation :
Solution :
Find
A
-1
....
A
-1
:
Formula :
Cramer’s rules
Assume :
Determinants :
x
1
:
x
2
:
x
3
:
Example :
Find x
1
and x
2
from these equation :
Solution :
x
1
and x
2
:
Proof :
Find x
1
, x
2
and x
3
from these equations :
Find Determinants :
Find x1, x2 and x3 :
Adjoint Matrix
Adjoint matrix of a square matrix is the transpose of
the matrix formed by cofactors of elements of
determinant |A|
How to calculate adjoint :
Calculate minor matrix for each element of matrix
Make cofactor matrix
cofactor is a sign minor, denoted by : C
ij
= (-1)
ij
. M
ij
Change to Transpose matrix.
Example
Find inverse for A :
Calculate |A| :
=(1.5.3 + 2.0.2 + 3.0.4) – (3.5.2 +1.0.4 + 2.0.3)
= 15 – 30
= - 15
Make a new matrix with minor and cofactor
→
Transpose that matrix :
Find x
1
, x
2
and x
3
from these equations :
Matrix form :
Formula :
Determinants :
Use minor cofactor :
New matrix K :
A
-1
:
Formula :
Questions
Find x1 and x2 from these equations :
2x
1
+ x
2
– 4 = 0
x
1
– 3x
2
+ 5 = 0
Find x, y and z from these equations :
x + y + z
–
6 = 0
2x
–
z + 1 = 0
x
–
y + 2z
–
5 = 0
Explore the concepts of linear equations, matrix forms, determinants, and finding solutions for variables like x1, x2, x3. Learn about Cramer's Rules, Adjoint Matrix, and calculating the inverse of a matrix through examples and formulas.
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Presentation Transcript
Example : Find x1and x2from these equation : Solution :
General form : Where : a1, a2, a3.. anand b are constantas x1, x2, x3.. xnare variables
Linear Equation in matrix form If we have some equations : Then, we can write :
General form : Where : Or, we can write :
Example : Find x1and x2from these equation : Solution : Find A-1....
A-1 : Formula :
Cramers rules Assume : Determinants :
x1 : x2 : x3 :
Example : Find x1 and x2 from these equation : Solution :
x1 and x2 : Proof :
Adjoint Matrix Adjoint matrix of a square matrix is the transpose of the matrix formed by cofactors of elements of determinant |A| How to calculate adjoint : Calculate minor matrix for each element of matrix Make cofactor matrix cofactor is a sign minor, denoted by : Cij = (-1)ij . Mij Change to Transpose matrix.
Example Find inverse for A : Calculate |A| : =(1.5.3 + 2.0.2 + 3.0.4) (3.5.2 +1.0.4 + 2.0.3) = 15 30 = - 15
Make a new matrix with minor and cofactor Transpose that matrix :
Matrix form : Formula :
Determinants : Use minor cofactor :
New matrix K : A-1 :
Questions Find x1 and x2 from these equations : 2x1+ x2 4 = 0 x1 3x2+ 5 = 0 Find x, y and z from these equations : x + y + z 6 = 0 2x z + 1 = 0 x y + 2z 5 = 0
