1. Matrices

2. Element - each value in a matrix; either a number or a constant.
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: 3x2
Dimensions: 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. Examples: Find the values for x and y
* Since the matrices are equal, the corresponding elements are equal!
* Form two linear equations.
* Solve the system using substitution.

7. Set each element equal and solve!2.
Set each element equal and solve!

8. Matrix Operations
Addition
Subtraction
Multiplication
Inverse

9. Addition

10. 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

13. 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
= (a11 x a22) - (a21 x a12)
For
det(A) = (2x3) – (1x5) = 1

27. Inverse of 2 x 2 matrix
Swap elements a11 and a22
Thus
becomes

28. Inverse of 2 x 2 matrix
Change sign of a12 and a21
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