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MAT 1234 Calculus I

This lesson explores the fundamental algebraic operations of matrices, including addition, subtraction, and scalar multiplication, with clear examples. We discuss non-“term-by-term” operations like matrix multiplication and its significance in linear equations. Key concepts such as dimensions, notations, row and column vectors, and special cases of matrices are covered, alongside transformative insights into the representation of linear systems using matrix multiplication. Understand the identity matrix and intriguing properties of matrix products, enhancing your grasp of this vital mathematical tool.

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MAT 1234 Calculus I

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  1. MAT 1234Calculus I 9.5 The Algebra of Matrices http://myhome.spu.edu/lauw

  2. HW • WebAssign 9.5

  3. Preview • Look at the algebraic operations of matrices • “term-by-term” operations • Matrix Addition and Subtraction • Scalar Multiplication • Non-“term-by-term” operations • Matrix Multiplication

  4. Matrix • If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn.

  5. Notations • Matrix

  6. Notations • Matrix

  7. Special Cases • Row Vector • Column Vector

  8. Matrix Addition and Subtraction • Let A = [aij] and B = [bij] be mxn matrices • Sum: A + B = [aij+bij] • Difference: A-B = [aij-bij] • (Term-by term operations)

  9. Example 1

  10. Scalar Multiplication Let A = [aij] be a mxn matrix and c a scalar. • Scalar Product: cA=[caij]

  11. Example 2

  12. Matrix Multiplication • Define multiplications between 2 matrices • Not “term-by-term” operations

  13. Motivation • The LHS of the linear equation consists of two pieces of information: • coefficients: 2, -3, and 4 • variables: x, y, and z

  14. Motivation • Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors.

  15. Row-Column Product

  16. Example 3

  17. Matrix Multiplication

  18. Example 4

  19. Example 5 (a)

  20. Example 5 (b)

  21. Example 5 (c)

  22. Example 5 (d)

  23. Example 5 (e)

  24. Example 5 (f)

  25. Interesting Facts • The product of mxp and pxn matrices is a mxn matrix. • In general, AB and BA are not the same even if both products are defined. • AB=0 does not necessary imply A=0 or B=0. • Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity.

  26. Identity Matrix nxn Square Matrix

  27. Representation of Linear System by Matrix Multiplication

  28. Representation of Linear System by Matrix Multiplication

  29. Representation of Linear System by Matrix Multiplication

  30. Representation of Linear System by Matrix Multiplication

  31. Remark It would be nice if “division” can be defined such that: (9.6) Inverse

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