1. Matrices

2. Matrix - a rectangular array of variables or constants in horizontal rows and vertical columns enclosed in brackets. Element - each value in a matrix; either a number or a constant. Dimension - number of rows by number of columns of a matrix. **A matrix is named by its dimensions.

3. Examples: Find the dimensions of each matrix. Dimensions: 3x2Dimensions: 4x1 Dimensions: 2x4

4. Different types of Matrices Column Matrix - a matrix with only one column. Row Matrix - a matrix with only one row. Square Matrix - a matrix that has the same number of rows and columns.

5. Equal Matrices - two matrices that have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix. *The definition of equal matrices can be used to find values when elements of the matrices are algebraic expressions.

6. * Since the matrices are equal, the corresponding elements are equal! * Form two linear equations. * Solve the system using substitution. Examples: Find the values for x and y

7. Set each element equal and solve! 2.

8. Matrix Operations  Addition  Subtraction  Multiplication  Inverse

9. Addition

11. Addition Conformability To add two matrices A and B:  # of rows in A = # of rows in B  # of columns in A = # of columns in B

12. Subtraction

14. Subtraction Conformability  To subtract two matrices A and B:  # of rows in A = # of rows in B  # of columns in A = # of columns in B

15. Multiplication Conformability  Regular Multiplication  To multiply two matrices A and B:  # of columns in A = # of rows in B  Multiply: A (m x n) by B (n by p)

16. Multiplication General Formula

17. Multiplication I

18. Multiplication II

19. Multiplication III

20. Multiplication IV

21. Multiplication V

22. Multiplication VI

23. Multiplication VII

24. Inner Product of a Vector  (Column) Vector c (n x 1)

25. Outer Product of a Vector  (Column) vector c (n x 1)

26. Inverse of 2 x 2 matrix  Find the determinant = (a 11 x a 22 ) - (a 21 x a 12 ) For det(A) = (2x3) – (1x5) = 1

27. Inverse of 2 x 2 matrix  Swap elements a 11 and a 22 Thus becomes

28. Inverse of 2 x 2 matrix  Change sign of a 12 and a 21 Thus becomes

29. Inverse of 2 x 2 matrix  Divide every element by the determinant Thus becomes (luckily the determinant was 1)

30. Inverse of 2 x 2 matrix  Check results with A -1 A = I Thus equals