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CHAPTER 2

CHAPTER 2. MATRIX. Chapter outline. 2.1 Introduction 2.2 Types of Matrices 2.3 Determinants 2.4 The Inverse of a Square Matrix 2.5 Types of Solutions to Systems of Linear Equations 2.6 Solving Systems of Equations 2.7 Eigenvalues and Eigenvectors 2.8 Applications of Matrices.

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CHAPTER 2

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  1. CHAPTER 2 MATRIX

  2. Chapter outline • 2.1 Introduction • 2.2 Types of Matrices • 2.3 Determinants • 2.4 The Inverse of a Square Matrix • 2.5 Types of Solutions to Systems of Linear Equations • 2.6 Solving Systems of Equations • 2.7 Eigenvalues and Eigenvectors • 2.8 Applications of Matrices

  3. 2.1 introduction • Definition 2.1 • A matrix is a rectangular array of elements or entries involving m rows and n columns. • Definition 2.2 • Two matrices are said to be equal if m = r and n = s ( are same size ) then A = B, and corresponding elements throughout must also equal where • If are called the main diagonal of matrix A.

  4. Example 2.1 (Exercise 2.1 in TextBook): • Find the values for the variables so that the matrices in each exercise are equal. • i. • ii.

  5. Example 2.2 (Exercise 2.2 in Textbook): • Give the order of each matrix. • Identify or explain why identification is not possible.

  6. 2.2 types of matrices • Square Matrix • A square matrix is any of order matrix and has the same number of columns as rows. • Diagonal Matrix • An matrix is called a diagonal matrix if

  7. Example 2.3 (Exercise 2.3 in Textbook): • Determine the matrices A and B are diagonal or not. • i. • ii.

  8. Scalar Matrix • Scalar matrix- the diagonal elements are equal. • Identity Matrix • Identity matrix is called identity matrix with “1” on the main diagonal and “0”.

  9. Zero Matrices • Zero Matrices – contain only “0” elements. • Negative Matrix • A negative matrix of • Upper Triangular • Upper triangular – if every element leading diagonal is zero.

  10. Transpose Matrix

  11. Properties of Transposition Operation • Let A, B matrices (different order) and k. Then • Example (Exercise 2.5 in Textbook):

  12. Symmetric Matrix • Symmetric matrix – where the elements are obey the rule • Example (Exercise 2.6 in Textbook):

  13. Skew Symmetric Matrix • Skew symmetric matrix – • Example (Exercise 2.7 in Textbook):

  14. Row Echelon Form (REF)

  15. Example (Exercise 2.8 in Textbook): • Determine whether each matrix is in row echelon form.

  16. Reduced Row Echelon Form (RREF) • A matrix is said RREF if it satisfies the following properties: • Any rows consisting entirely of zeros occur at the bottom of the matrix. • For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). • For each non zero row, the number 1 appears to the right of the leading 1 of the previous row. • If a column contains a leading 1, then all other entries in the column are zero.

  17. Example: • Determine whether each matrix is in reduced row echelon form.

  18. Addition and Subtraction of Matrix • Two matrices can be addition and subtraction only if they are both in the same order. • Properties of Matrices Addition and Subtraction

  19. Example (Addition) : • 1. • 2. • 3.

  20. Example (Subtraction):

  21. Scalar Multiplication • Scalar multiplication is denoted • Properties of Scalar Multiplication

  22. Properties of Matrix Multiplication

  23. Example (Exercise 2.10 in Textbook): • 1.

  24. Exercise: • 1. • 2. • 3.

  25. 4. • 5. • 6. Find

  26. 2.3 Determinants • Second Order Determinants • Determinant of A denoted by • Example : • Evaluate the determinant of

  27. Third Order Determinants • Methods used to evaluate the determinants above is limited to only 2 X 2 and 3 X 3 matrices. Matrices with higher order can be solved using minor and cofactor methods.

  28. Example (Exercise 2.11 in Textbook): • Evaluate the determinant of matrix • Exercise : • Find the determinants for matrix

  29. Minors and Cofactors • Minor • Cofactor

  30. Example (Exercise 2.12 in Textbook) : • Find all the minors and cofactors of • Exercise :

  31. High Order Determinants

  32. Example ((Exercise 2.13 in Textbook) : • Find the determinant of • by expanding cofactors in the second row.

  33. Exercise: • Find the determinant of by expansion • the second column.

  34. Adjoint • Example (Exercise 2.14 in Textbook) : • Find the adjoint of the matrix

  35. Exercise : • Find the adjoint of the matrix

  36. The inverse of a square matrix • Inverse of a Matrix

  37. Example (Exercise 2.15 of Textbook) :

  38. Finding the Inverse of a 2 by 2 Matrix • Theorem 1

  39. Example (Exercise 2.16 of Textbook) :

  40. Finding the Inverse of a 3 by 3 or Higher Matrix by Using Cofactor Method

  41. Example (Exercise 2.17 Of Textbook) :

  42. Finding the Inverse of a 3 by 3 or Higher Matrix by Elementary Row Operation • Characteristics of Elementary Row Operations (ERO)

  43. Example (Exercise 2.18 of Textbook) :

  44. Exercise :

  45. Systems of linear equations • Types of Solutions • Solving Systems of Equations • Eigenvalue and Eigenvectors • Types of Solutions • System with Unique Solution (Independent) • A system which has unique solution. • Can find the values of

  46. A system with Infinitely Many Solutions (Dependent) • A system has infinitely many solutions • Row Echelon Form (REF) has a row of the form • In general whatever value of , the equation is satisfied if . So we define a free variable, s.

  47. System with No Solution (Inconsistent) • A system has no solution. • REF has a row of the form c is a constant.

  48. Inverses of Matrices

  49. When A is a square matrix. Note that A and B are matrices with numerical elements. To an expression for the unknowns, that is the elements of X. • Premultiplying both sides of the equation by the inverse of A, if it exists, to obtain • The left hand side can be simplified by noting that multiplying a matrix by its inverse gives the identity matrix, that is • Hence , • Left hand side simplifies to

  50. Example (Exercise 2.19 of Textbook): • Solve the following system of linear equations using the inverse matrix. • i. • ii.

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