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Operations with Matrices: Addition, Subtraction, and Multiplication in Linear Algebra

This section covers fundamental operations involving matrices, including addition, subtraction, scalar multiplication, and matrix multiplication. Students will learn how to perform term-by-term operations with examples illustrating the sum and difference of matrices. Additionally, the discussion includes non-term-by-term operations, particularly focusing on defining matrix multiplication and its implications in solving linear equations. Understanding these concepts lays the groundwork for representing linear systems and manipulating matrices in subsequent topics.

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Operations with Matrices: Addition, Subtraction, and Multiplication in Linear Algebra

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  1. MAT 2401Linear Algebra 2.1 Operations with Matrices http://myhome.spu.edu/lauw

  2. Today • WebAssign 2.1 • Written HW • Again, today may be longer. It is more efficient to bundle together some materials from 2.2. • Next class session will be shorter.

  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. Zero Matrix mxn Matrix with all zero entries

  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. Representation of Linear System by Matrix Multiplication

  32. Remark It would be nice if “division” can be defined such that: (2.3) Inverse

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