1. LINEAR ALGEBRA

2. Example :
Find x1 and x2 from these equation :
Solution :

3. General form : Where : a1, a2, a3 .. an and b are constantas
x1, x2, x3 .. xn are variables

4. Linear Equation in matrix form
If we have some equations :
Then, we can write :

5. General form :
Where :
Or, we can write :

6. Example : Find x1 and x2 from these equation : Solution :
Find A

7. A-1 :
Formula :

8. Cramer’s rules
Assume :
Determinants :

9. x1 :
x2 :
x3 :

10. Example :
Find x1 and x2 from these equation :
Solution :

11. x1 and x2 :
Proof :

12. Find x1, x2 and x3 from these equations :

13. Find Determinants :

14. Find x1, x2 and x3 :

15. 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.

16. Example Find inverse for A : Calculate |A| :
=( ) – ( )
= 15 – 30
= - 15

17. Make a new matrix with minor and cofactor→
Transpose that matrix :

18. Find x1, x2 and x3 from these equations :

19. Matrix form :
Formula :

20. Determinants :
Use minor cofactor :

22. New matrix K :
A-1 :

23. Formula :

24. Questions x + y + z – 6 = 0 2x – z + 1 = 0 x – y + 2z – 5 = 0
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