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Matrices
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Matrices

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For those that are learning how to use matrices or just need a quick refresher

For those that are learning how to use matrices or just need a quick refresher

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Matrices Presentation Transcript

  • 1. Presented by: Katie Arnold
    MATRIX OPERATIONS
  • 2. Preview of things to come
    • We will begin reviewing what we know about matrices
    • 3. We will then learn about
    matrix addition
    matrix subtraction
    matrix multiplication
    determinants of 2x2 and 3x3 matrices
  • 4. Introduction to Matrices
    • A matrix (plural: matrices) is a rectangular array of numbers.
    • 5. Each of these numbers is called an entry or an element.
    • 6. The elements of a matrix usually are enclosed by two brackets. Here are some examples.
    , , ,
  • 7. Introduction to Matrices (continued)
    • The horizontal lines of a matrix are called rows and the vertical lines are called columns.
    • 8. The size of a matrix is identified by the number of its rows and columns.
    • 9. The size of a matrix is denoted as an index for the label of the matrix. The index usually is in the form of m × n in which m is the number of rows and n is the number of columns.
  • Introduction to Matrices (continued)
    • For example, S4 × 3 represents a matrix S, which has four rows and three columns.
    • 10. Each element or entry in a matrix is identified by its location. The location of an entry is the point in which the row and the column to which the entry belongs intersect. 
    • 11. Determine the size of the matrix:
    answer
    2x4
  • 12. ADDING MATRICES
    • Adding two or more matrices is a simple task. They just must have the same size. Then, adding the corresponding entries of the given matrices results in the addition of the matrices.
    • 13. For example, to add the matrices and compute the following:
    • 14. Sum of entries in first rows and first columns = 3 + 1 = 4
    • 15. Sum of the entries in first rows and second columns = 8 + 0 = 8
    • 16. Sum of entries in second rows and first columns = 2 + 2 = 4
    • 17. Sum of entries in second rows and second columns = 11 + 3 = 14 
    Now, we have all the entries of the resulting matrix. So,
  • 18. SUBTRACTING MATRICES
    • Subtracting two matrices is also a simple task. They just must have the same size. Then, subtracting the corresponding entries of the given matrices results in the addition of the matrices.
    • 19. For example, to add the matrices and compute the following:
    • 20. Difference of entries in first rows and first columns = 3 - 1 = 2
    • 21. Difference of the entries in first rows and second columns = 8 - 0 = 8
    • 22. Difference of entries in second rows and first columns = 2 - 2 = 0
    • 23. Difference of entries in second rows and second columns = 11 - 3 = 8 
    Now, we have all the entries of the resulting matrix. So,
  • 24. MULTIPLYING 2x2 MATRICES
    • Two matrices A and B can only be multiplied if the number of columns of A is equal to the number of rows of B.
    • 25. In that case the product Amxnx Bnxr will result in a matrix of size mxr
    • 26. Let ; we are trying to compute
    In this case, the coefficients of matrix C will be computed as follows:
  • 27. MULTIPLYING 2x2 MATRICES (continued)
    • Example
    Then
    It must be mentioned that and this can easily be checked.
  • 28. DETERMINANTS
    • Only square matrices have determinants (m≠n)
    • 29. The determinant of a matrix is shown by placing the entries of the matrix between two vertical bars | |
    • 30. For example the determinants of matrix
    is
  • 31. THE DETERMINANT OF A 2x2 MATRIX
    • For a 2x2 square matrix the determinant is defined as below:
    • 32. For example,
  • THE DETERMINANT OF A 3x3 MATRIX
    • For a 2x2 square matrix the determinant is defined as below:
    • 33. For example,
  • Homework Problems
    Compute the following: