1. Lecture 8 System of Linear Differential Equations fall semester
Instructor: A. S. Brwa / MSc. In Structural Engineering
College of Engineering / Ishik University

2. Ishik University
Review of Matrix
Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. Matrices with more than one row and one column denoted by bold, upper case letters (e.g., X).
𝟏 𝟑 𝟎 −𝟐 𝟔 𝟒
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

3. Review of Matrix Look at this Matrix:
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Review of Matrix
Look at this Matrix:
These are called elements or entries.
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

4. Review of Matrix Ishik University
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

5. Review of Matrix What's the size of this matrix?
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Review of Matrix
What's the size of this matrix?
3 rows and 2 columns. This is a 3 by 2 matrix (3x2).
Note: A 2 x 3 matrix and a 3 x 2 matrix are definitely different sizes!
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

6. Ishik University
Review of Matrix
Sometimes, we'll need to refer to a specific entry, so we have a special "tagging" system.  It's based on rows and columns:
This element is located in the row and in the column
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

7. Review of Matrix Ishik University
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

8. Ishik University
Review of Matrix
A vector: is a special type of matrix that has only one row (called a row vector) or one column (called a column vector). Below, a is a column vector while b is a row vector.
Vectors are denoted by bold, lower case letters (e.g., x).
Row vector
Column vector
A scalar: is a matrix with only one row and one column. It is customary to denote scalars by italicized, lower case letters (e.g., x)
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

9. Ishik University
Review of Matrix
Square Matrices: A square matrix A has the same number of rows and columns. That is, A is n x n. In this case, A is said to have order n.
Faculty of Engineering – Differential Equations – Lecture 8– System of Linear Differential Equations

10. Ishik University
Review of Matrix
The Zero Matrix: The zero matrix is defined to be 0 = (0), whose dimensions depend on the context. For example,
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

11. Ishik University
Review of Matrix
Identity matrix: is a diagonal matrix with 1’s and only 1’s on the diagonal and zeros everywhere else. The identity matrix is almost always denoted as I.
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

12. Review of Matrix Adding and Subtracting Matrices\
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Review of Matrix
Adding and Subtracting Matrices\
Example: Subtract Matrix A and B
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

13. Review of Matrix Adding and Subtracting Matrices\
Ishik University
Review of Matrix
Adding and Subtracting Matrices\
Adding and subtracting matrix is really easy, but THE SIZES MUST BE THE SAME!
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

14. Review of Matrix Adding and Subtracting Matrices\
Ishik University
Review of Matrix
Adding and Subtracting Matrices\
Adding and subtracting matrix is really easy, but THE SIZES MUST BE THE SAME!
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

15. Review of Matrix Adding and Subtracting Matrices\
Ishik University
Review of Matrix
Adding and Subtracting Matrices\
Example: Add Matrix A and B
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

16. Review of Matrix Multiplication of Matrices\
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Review of Matrix
Multiplication of Matrices\
Multiplication by scalar, A scalar is just a number like 3 or -5 or  2 7   or .4. We can multiply a matrix by some value:
𝑺𝒄𝒂𝒍𝒂𝒓(𝑎)= 𝟑 𝐚𝐧𝐝
Matrix 𝐀= 𝟏 𝟑 𝟎 −𝟐 𝟔 𝟒
𝑎∗𝑨=𝟑 𝟏 𝟑 𝟎 −𝟐 𝟔 𝟒
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

17. Review of Matrix Multiplication of Matrices\
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication by scalar. Example: If Matrix
Find 2A... (2 times A)
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

18. Review of Matrix Multiplication of Matrices\
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication by scalar.  Example: 𝐼𝑓 𝑀𝑎𝑡𝑟𝑖𝑥 𝐴= 𝟏 𝟒 𝟐 𝟓 𝟑 𝟖
Find 5A.
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

19. Review of Matrix Multiplication of Matrices\
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication by scalar.  Example: Now, let’s try more difficult one 𝑰𝒇 𝑴𝒂𝒕𝒓𝒊𝒙 𝑨= 𝟏 𝟒 𝟐 𝟓 𝟑 𝟖 𝒂𝒏𝒅 𝑩= 𝟒 𝟖 𝟕 𝟓 𝟓 𝟏𝟏 , 𝐟𝐢𝐧𝐝 𝟓𝐀−𝐁.
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

20. Review of Matrix Multiplication of Matrices\
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication by scalar.  Example: Now, let’s try more difficult one 𝑰𝒇 𝑴𝒂𝒕𝒓𝒊𝒙 𝑨= 𝟏 𝟒 𝟐 𝟓 𝟑 𝟖 𝒂𝒏𝒅 𝑩= 𝟒 𝟖 𝟕 𝟓 𝟓 𝟏𝟏 , 𝐟𝐢𝐧𝐝 𝟓𝐀−𝐁.
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

21. And the answer will be a 3 x 1 matrix
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication of two matrices.  To multiply two matrices together is a bit more difficult. First of all, the size of the two matrices you are multiplying is super important!
If the size of matrix A is 3 x 4 and the size of matrix B is 4 x 1
And the answer will be a 3 x 1 matrix
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

22. Review of Matrix Multiplication of Matrices\
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication of two matrices.  
Example: If Matrix 𝑨= −𝟏 𝟓 𝟐 −𝟑 𝒂𝒏𝒅 𝑩 𝟐 𝟑 𝟓 𝟏 𝑭𝒊𝒏𝒅 𝑨∙𝑩
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

23. Kurdistan Technical Institute (KTI)
Ishik University
Review of Matrix
𝑨= −𝟏 𝟓 𝟐 −𝟑 𝒂𝒏𝒅 𝑩 𝟐 𝟑 𝟓 𝟏 𝑭𝒊𝒏𝒅 𝑨∙𝑩
A is 2 by 2 and B is 2 by 2. So, the result is
The result is 2 by 2
23/27
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations
Department of Petroleum Refining

24. Kurdistan Technical Institute (KTI)
Ishik University
Review of Matrix
𝑨= −𝟏 𝟓 𝟐 −𝟑 𝒂𝒏𝒅 𝑩 𝟐 𝟑 𝟓 𝟏 𝑭𝒊𝒏𝒅 𝑨∙𝑩
A is 2 by 2 and B is 2 by 2. So, the result is
The result is 2 by 2
2 rows and 2 columns
24/27
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations
Department of Petroleum Refining

25. Review of Matrix Multiplication of Matrices\
Ishik University
Review of Matrix
Multiplication of Matrices\
Multiplication of two matrices.  
Example: If Matrix𝑨= 𝟎 𝟗 𝟖 −𝟔 𝟒 𝟕 𝒂𝒏𝒅 𝑩 −𝟏 𝟓 𝟐 −𝟑 𝑭𝒊𝒏𝒅 𝑨∙𝑩
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

26. Kurdistan Technical Institute (KTI)
Ishik University
Review of Matrix
𝑨= 𝟎 𝟗 𝟖 −𝟔 𝟒 𝟕 𝒂𝒏𝒅 𝑩 −𝟏 𝟓 𝟐 −𝟑 𝑭𝒊𝒏𝒅 𝑨∙𝑩
26/27
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations
Department of Petroleum Refining

27. Kurdistan Technical Institute (KTI)
Ishik University
Review of Matrix
𝑨= 𝟎 𝟗 𝟖 −𝟔 𝟒 𝟕 𝒂𝒏𝒅 𝑩 −𝟏 𝟓 𝟐 −𝟑 𝑭𝒊𝒏𝒅 𝑨∙𝑩
27/27
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations
Department of Petroleum Refining

28. Ishik University
Review of Matrix
Determinant Matrix The determinant of a matrix is a special number that can be calculated from a square matrix. Its very helpful in solving the system of linear equations and inverse matrix
Example: If matrix
Then, its determinate |𝑨| = 𝑎𝑑 −𝑏𝑐
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

29. Review of Matrix Determinant Matrix Example: If matrix A =
Ishik University
Review of Matrix
Determinant Matrix
Example: If matrix A =
Then, its determinate |A| = 3×6 − 8×4 = 18 − 32 = −14
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

30. Review of Matrix Determinant Matrix Example: If matrix
Ishik University
Review of Matrix
Determinant Matrix
Example: If matrix
Then, its determinate 𝑩 = 𝟒×𝟖 − 𝟔×𝟑 = 𝟑𝟐−𝟏𝟖 = 𝟏𝟒
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

31. Review of Matrix Determinant Matrix Example: If matrix
Ishik University
Review of Matrix
Determinant Matrix
Example: If matrix
The determinant is;
|𝑨| = 𝒂(𝒆𝒊 − 𝒇𝒉) − 𝒃(𝒅𝒊 − 𝒇𝒈) + 𝒄(𝒅𝒉 − 𝒆𝒈)
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

32. Review of Matrix Determinant Matrix Example: If matrix
Ishik University
Review of Matrix
Determinant Matrix
Example: If matrix
Then, its determinate 𝑪 =
6× −2×7 − 5×8 − 1× 4×7 − 5×2 +1× 4×8 − −2×2 = 6× −54 − 1× × 36 = −𝟑𝟎𝟔
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

33. Review of Matrix Inverse Matrix Example: If matrix Find Inverse of A
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Review of Matrix
Inverse Matrix
Example: If matrix
Find Inverse of A
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

34. Review of Matrix Inverse Matrix Example: If matrix
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Review of Matrix
Inverse Matrix
Example: If matrix
First need to find determinants of A
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

35. Review of Matrix Inverse Matrix First need to find determinants of A
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Review of Matrix
Inverse Matrix
First need to find determinants of A
If determinant of A is NOT zero, then matrix A is invertible (non-singular).
If determinant of A is zero, then matrix A is NOT invertible (singular).
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

36. Review of Matrix Second need to find matrix of cofactors
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Review of Matrix
Second need to find matrix of cofactors
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

37. Review of Matrix Third Step: Adjugate (also called Adjoint)
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Review of Matrix
Third Step: Adjugate (also called Adjoint)
Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same):
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations

38. Review of Matrix Forth Step: Multiply by 1/Determinant
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Review of Matrix
Forth Step: Multiply by 1/Determinant
Faculty of Engineering – Differential Equations – Lecture 8 – System of Linear Differential Equations