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Topic 2 MATRICES & systems of linear equations

Topic 2 MATRICES & systems of linear equations. LEARNING OUTCOMES. At the end of this topic, student should be able to : D efination of matrix

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Topic 2 MATRICES & systems of linear equations

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  1. Topic 2 MATRICES & systems of linear equations

  2. LEARNING OUTCOMES At the end of this topic, student should be able to : • Defination of matrix • Identify the different types of matrices such as rectangular, column, row, square , zero / null , diagonal, scalar, upper triangular, lower triangular and identity matrices • Solve the equality of matrices • Perform operations on matrices such as addition, subtraction, scalar multiplication of two matrices • Identify Transpose of Matrix • Define the determinant of matrix and find the determinant of 2 x 2 and 3 x 3 matrix • Find minor and cofactor of 2 x 2 and 3 x 3 matrix • Define Inverse Matrix • Find the Inverse Matrix by using Adjoint Matrix • Write a system of linear equations • Solve the system of linear equations by using Inverse Matrix and Cramer’s Rule

  3. Definition of Matrix A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called elements or the entries of the matrix.

  4. Example of a matrix A = Order of a matrix = row x column = (m x n) Order of A matrix = 3 x 3 Element of A matrix = 1,2,3,4,5,6,7,8,9

  5. Types of Matrices 1. Rectangular Matrix A = 2 x 4 2. Column Matrix A = 2 x 1

  6. 3. Row Matrix A = 1 x 3 4. Square Matrix A = 2 x 2 5. Diagonal Matrix A = 3 x 3

  7. 6. Zero / Null Matrix A= 2 x 2 7. Scalar Matrix A = 3 x 3 8. Upper Triangular Matrix A = 3 x 3

  8. 9. Lower Triangular Matrix A = 3 x 3 10. Identity Matrix A = 3 x 3 OR A = 2 x 2

  9. Equality of Matrices Two matrices are equal if they have the same order and same entries.

  10. Exercises 1. Find the value of x and y for the following : (a) (b) (c) (d)

  11. Operations on Matrices Additions / Subtractions Additions or subtractions of matrices can be done if they have the same dimensions whereby the two matrices must have the same number of rows and the same number of columns. When two matrices are addedor subtracted then the order of matrix should be the same.

  12. Example : A + B = C

  13. Multiplication Scalar Multiplication Example : A = 2A = =

  14. Multiplication of Two Matrices Necessary condition for matrix multiplication • Column of first matrix should be equal to the row of the second of matrix. Example :

  15. Exercises 1. A =B =C = Find : (a) A + C (b) C – A (c) 3A – 2C (d) A + 2C (e) AB

  16. Transpose of a Matrix A matrix obtain by interchangingthe rows and and columns of the original matrix. It is denoted by or . If is an is an matrix, that is the matrix.

  17. Example : A =

  18. Determinant of Matrix The determinant of matrix is a unique real number for every square matrix.The determinant of a square matrix is denoted by Det A or . Determinant of Matrix 2 x 2 Let us consider a 2 x 2 matrix :

  19. Example : Find the value of the determinant for matrix A. Solution :

  20. Determinant for Matrix 3 x 3 Let us consider a 3 x 3 matrix :

  21. Example : Given , find or determinant of A. Solution : =

  22. Minor of 2 x 2 Matrix Let us consider matrix 2 x 2,

  23. Minor of 3 x 3 Matrix Let us consider matrix 3 x 3,

  24. Example : Find and . Solution: (a) (b)

  25. Cofactor Cofactor for 2 x 2 Matrix Let us consider matrix 2 x2

  26. Example : Given , find the cofactor for A. Solution :

  27. Cofactorfor 3 x 3 Matrix Let us consider matrix 3 x 3 , =

  28. Example : Given , find cofactor for A. Solution :

  29. Inverse Matrix by using Adjoint Matrix where Steps : 1. Find 2. Identify 3. Identify 4. Substitute and Adj A in Inverse Matrix formulae.

  30. Example 1 : Find if . Solution: Step 1 : Step 2 : Step 3 : Step 4 :

  31. Example 2 : Find if . Solution: Step 1 : Step 2 : Step 3 : Step 4 :

  32. Systems of Linear Equations A system of linear equations is a collection of two @ more linear equations, each containing one or more variables. The following is a system of three equations containing three variables. Using a matrix notation, we can write this system in simplified form. This is called the augmented matrix of the system.

  33. Exercise Write the augmented matrix of each system. (a) (b)

  34. Solving a system using an Inverse Matrix Consider the pair of simultenous equations Let the matrix of coefficient be , that is In matrix form of system above can be written as

  35. Example Solve the following equations by using Inverse Matrix. Solution Step 1: Step 5 : Step 2 : Step 3 : Step 4 :

  36. (b) Solving a system using Cramer’s Rule Consider the pair of simultenous equations Let the matrix of coefficient be , that is Therefore by using Cramer’s Rule for 2 x 2 Matrix

  37. Solve the system by using Cramer’s Rule • 8x+5y=2 • 2x-4y=-10 Example :

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