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Matrices & Systems of Linear Equations

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  1. Matrices & Systems of Linear Equations

  2. Special Matrices

  3. Special Matrices

  4. Equality of Matrices Two matrices are said to be equal if they have the same size and their corresponding entries are equal

  5. Equality of Matrices Use the given equality to find x, y and z

  6. Matrix Addition and SubtractionExample (1)

  7. Matrix Addition and SubtractionExample (2)

  8. Multiplication of a Matrix by a Scalar

  9. Matrix Multiplication(n by m) Matrix X (m by k) MatrixThe number of columns of the matrix on the left= number of rows of the matrix on the right The result is a (n by k) Matrix

  10. Matrix Multiplication3x3 X 3x3

  11. Matrix Multiplication1x3 X 3x3→ 1x3

  12. Example (1)

  13. Example (2)(1X3) X (3X3) → 1X3

  14. Example (3)(3X1) X (1X2) → 3X2

  15. Example (4)

  16. Transpose of Matrix

  17. Properties of the Transpose

  18. Matrix ReductionDefinitions (1) 1. Zero Row:A row consisting entirely of zeros 2. Nonzero Row:A row having at least one nonzero entry 3. Leading Entry of a row:The first nonzero entry of a row.

  19. Matrix ReductionDefinitions (2) Reduced Matrix: A matrix satisfying the following: 1. All zero rows, if any, are at the bottom of the matrix 2. The leading entry of a row is 1 3. All other entries in the column in which the leading entry is located are zeros. 4. A leading entry in a row is to the right of a leading entry in any row above it.

  20. Examples of Reduced Matrices

  21. Examples matrices that are not reduced

  22. Elementary Row Operations 1. Interchanging two rows 2. Replacing a row by a nonzero multiple of itself 3. Replacing a row by the sum of that row and a nonzero multiple of another row.

  23. Interchanging Rows

  24. Replacing a row by a nonzero multiple of itself

  25. Replacing a row by the sum of that row and a nonzero multiple of another row

  26. Augmented Matrix Representing a System of linear Equations

  27. Solving a System of Linear Equations by Reducing its Augmented Matrix Using Row Operations

  28. Solution

  29. Solution of the System

  30. The Idea behind the Reduction Method

  31. Interchanging the First & the Second Row

  32. Multiplying the first Equation by 1/3

  33. Subtracting from the Third Equation 5 times the First Equation

  34. Subtracting from the First Equation 2 times the Second Equation

  35. Adding to the Third Equation 12 times the Second Equation

  36. Dividing the Third Equation by 40

  37. Adding to the First Equation 7 times the third Equation

  38. Subtracting from the Second Equation 3/2 times the third Equation

  39. Systems with infinitely many Solutions

  40. Systems with infinitely many Solutions