Lesson 1 - Introduction to Matrices
- 2. Learning Intention and Success Criteria ļ Learning Intention: Students will understand the meaning of a variety of terms relating to matrices ļ Success Criteria: You can use accurate mathematical terminology to describe matrices and correctly identify characteristics of a variety of matrices.
- 3. What is a Matrix? ļ Matrix: A two- dimensional array with rows and columns used to organise data. ļ Denoted with a capital letter ļ Has square brackets around it ļ The data is often numerical, but does not have to be
- 4. Why Matrices? ļ Simply put, matrices are a simple way of displaying data, with extraneous information removed ļ Other applications: ļ Graphic design (like reflections and shadows) ļ Solving Equations
- 5. Vocabulary ļ Order: The dimension of the matrix "row x column" m x n ļ The order of matrix š“ is 2 Ć 3 because there are 2 rows and 3 columns ļ š: The number of rows (horizontal) ļ š: The number of columns (vertical) ļ (Do not use M or N as names of matrices) ļ Element: the term for each number in a matrix ļ Referring to entries in a matrix: ļ Lowercase letter (the same letter as the name of the matrix) ļ Followed by numbers for row,column ļ Ex: š1,2 would be the entry in the first row and the second column of the matrix š“.
- 6. Example #1 For the matrix š“ = 2 ā5 3 4 0 8 , a) State the order ļ Matrix š“ has order 2 Ć 3 b) State the name of the matrix ļ š“ c) State the values of the elements in position: i. š2,1 ii. š1,3 d) State the position of the element -5. ļ 1st row, 2nd column: the position is š1,2 ļ 2nd row, 1st column = 4 ļ 1st row, 3rd column = 3
- 7. Example #2 B is a 2 by 3 matrix. The elements of the matrix are defined according to the rule ššš = š ā š. What is the matrix? š1,1 = 1 ā 1 = 0 š2,1 = 2 ā 1 = 1 Calculate this for each element of šµ šµ = 0 ā1 ā2 1 0 ā1
- 8. Special Types of Matrices ļ Column Matrix: has exactly one column. Dimension is š Ć 1 ļ Row Matrix: has exactly one row. Dimension is 1 Ć š ļ Square matrix: has the same number of rows and columns. Dimension is š Ć š ļ Transpose of a Matrix: Given a matrix š“, the transpose of A (denoted š“ š) swaps the rows and columns ļ Entry š1,2 in matrix š“ will become entry š2,1 š in matrix š“ š ļ If š“ has dimension š Ć š, then š“ š has dimensions š Ć š ļ E.g. If š“ = 1 3 5 ā7 ā8 ā9 , then š“ š = 1 ā7 3 ā8 5 ā9
- 9. Square Matrices Sub-categories ļ Leading diagonal: The entries going from the top left to the bottom right of a square matrix ļ Diagonal Matrix: All entries not on the leading diagonal must be zero. ļ šµ = 3 0 0 0 5 0 0 0 ā4 ļ Identity Matrix (named I): A diagonal matrix where the leading diagonal is filled with 1s ļ š¼ = 1 0 0 0 1 0 0 0 1 Triangular Matrix ļ Upper-triangular: when all entries below the leading diagonal are zero. ļ 1 2 0 0 3 4 0 0 5 ļ Lower-triangular: when all entries above the leading diagonal are zero. ļ 1 0 0 0 ā4 0 3 7 1 ļ Note that the all diagonal matrices are both upper and lower- triangular