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# Matrices

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

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### Matrices

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