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Matrices Fundamentals: Types, Operations & Forms
- 1. Matrices MM1 module 3, lecture 1
- 2. Slide number 2 Matrices lesson 1 objectives • After this lesson you should have a clear understanding of: • What matrices are; • Performing basic operations on matrices; • Special forms of matrices
- 3. Slide number 3 What is a matrix? • A matrix is a rectangular array of elements • The elements may be of any type (e.g. integer, real, complex, logical, or even other matrices). • In this course we will only consider matrices that have integer or real number values. −5 0 1 2 3 −4 −9 2 3 1 4 2
- 4. Slide number 4 Order of matrices… • Order 4 × 3: • Order 3 × 4: 3 columns→ 4 rows ↓ −5 0 1 2 3 −4 −9 2 6 3 1 4 4 columns→ 3 rows ↓ −5 0 1 2 3 −4 −9 2 3 1 4 2
- 5. Slide number 5 …Order of matrices • Order 2 × 4: • Order 1 × 6: • Order 3 × 1: 1 columns→ 3 rows ↓ 3 −1 5 6 columns→ 1 rows ↓ 2 1 1 −2 1 −5[ ] 4 columns→ 2 rows ↓ −23 −0.5 4.3 12 8 2 8 1
- 6. Slide number 6 Specifying matrix elements • aij denotes the element of the matrix A on the ith row and jth column. A= column j → row i ↓ −5 0 1 2 3 −4 −9 2 6 3 1 4 • a12 = 0 • a21 = 2 • a23 = -4 • a32 = 2 • a41 = 3 • a43 = 4
- 7. Slide number 7 Matrix operations: scalar multiplication • Multiplying an m × n matrix by a scalar results in an m × n matrix with each of its elements multiplied by the scalar. • e.g. −−− −− −− − = − − − − − − − = − − − 826 12418 864 2010 413 629 432 105 2 1239 18627 1296 3015 413 629 432 105 3
- 8. Slide number 8 Matrix operations: addition… • Adding or subtracting an m × n matrix by an m × n matrix results in an m × n matrix with each of its elements added or subtracted. • e.g. − − − −−− = − − −− − − − − − − − − = − − −− + − − − 431 8112 113 136 024 213 541 231 413 629 432 105 417 436 971 334 024 213 541 231 413 629 432 105
- 9. Slide number 9 …Matrix operations: addition • Note that matrices being added or subtracted must be of the same order. • e.g. invalid! 113 201 413 629 432 105 = − + − − −
- 10. Slide number 10 Special matrices: row and column • A 1 × n matrix is called a row matrix. e.g. • An m × 1 matrix is called a column matrix. e.g. 1 columns→ 3 rows ↓ 3 −1 5 6 columns→ 1 rows ↓ 2 1 1 −2 1 −5[ ]
- 11. Slide number 11 Special matrices: square • An n × n matrix is called a square matrix. i.e. a square matrix has the same number of rows and columns. e.g. [ ] −− −− −− − 4705 7350 0523 5031 211 010 210 21 32 1
- 12. Slide number 12 Special matrices: distance • A distance matrix is a square matrix that shows the distances between locations. Distance matrices are frequently used on road maps to show the number of kilometres between major cities. Note that the main diagonal elements are all zeros.
- 13. Slide number 13 Matrices lesson 1 objectives • After this lesson you should have a clear understanding of: • What matrices are; • Performing basic operations on matrices; • Special forms of matrices;