1. Matrix Algebra Basics Chapter 3 Section 5

2. Algebra

3. Matrix A matrix is any doubly subscripted array of elements arranged in rows and columns.

4. Row Matrix [1 x n] matrix

5. Column Matrix [m x 1] matrix

6. Square Matrix Same number of rows and columns

7. The Identity

8. Identity Matrix Square matrix with ones on the diagonal and zeros elsewhere.

9. Transpose Matrix Rows become columns and columns become rows

10. Matrix Addition and Subtraction A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: Note: all three matrices are of the same dimension

11. Addition IfIf and then

12. Matrix Addition Example

13. Matrix addition: Ex 2: (Matrix addition)

14. Matrix Subtraction C = A - B Is defined by

15. Matrix subtraction: Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3A, (b) –B, (c) 3A – B

16. (a)(a) (b) (b) (c)(c) Solution:

17. Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A = cA + dA Properties of matrix addition and scalar multiplication:

18. Matrix Multiplication Matrices A and B have these dimensions: [r x c] and [s x d]

19. Matrix Multiplication Matrices A and B can be multiplied if: [r x c] and [s x d] c = s

20. Matrix Multiplication The resulting matrix will have the dimensions: [r x c] and [s x d] r x d

21. Computation: A x B = C [2 x 2] [2 x 3]

23. Computation: A x B = C [3 x 2][2 x 3] A and B can be multiplied [3 x 3]

24. Computation: A x B = C [3 x 2][2 x 3] [3 x 3] Result is 3 x 3

25. Inversion

26. Matrix Inversion Like a reciprocal in scalar math Like the number one in scalar math

27. Linear System of Simultaneous Equations First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct: 9 arrests - there were twice as many felonies as the first precinct.

28. Solution Note: Inverse ofis Premultiply both sides by inverse matrix A square matrix multiplied by its inverse results in the identity matrix. A 2x2 identity matrix multiplied by the 2x1 matrix results in the original 2x1 matrix.

29. n equations in n variables: unknown values of x can be found using the inverse of matrix A such that General Form

30. Garin-Lowry Model The object is to find x given A and y. This is done by solving for x :

31. Matrix Operations in Excel Select the cells in which the answer will appear

32. Matrix Multiplication in Excel 1)Enter “=mmult(“ 2)Select the cells of the first matrix 3)Enter comma “,” 4)Select the cells of the second matrix 5)Enter “)”

33. Matrix Multiplication in Excel Enter these three key strokes at the same time: control shift enter

34. Matrix Inversion in Excel Follow the same procedure Select cells in which answer is to be displayed Enter the formula: =minverse( Select the cells containing the matrix to be inverted Close parenthesis – type “)” Press three keys: Control, shift, enter