1. STROUD Worked examples and exercises are in the text PROGRAMME 5 MATRICES

2. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

3. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

4. STROUD Worked examples and exercises are in the text Matrices – definitions Programme 5: Matrices A matrix is a set of real or complex numbers (called elements) arranged in rows and columns to form a rectangular array. A matrix having m rows and n columns is called an m × n matrix. For example: is a 2 × 3 matrix.

5. STROUD Worked examples and exercises are in the text Matrices – definitions Programme 5: Matrices Row matrix A row matrix consists of a single row. For example: Column matrix A column matrix consists of a single column. For example:

6. STROUD Worked examples and exercises are in the text Matrices – definitions Programme 5: Matrices Double suffix notation Each element of a matrix has its own address denoted by double suffices, the first indicating the row and the second indicating the column. For example, the elements of 3 × 4 matrix can be written as:

7. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

8. STROUD Worked examples and exercises are in the text Matrix notation Programme 5: Matrices Where there is no ambiguity a matrix can be represented by a single general element in brackets or by a capital letter in bold type.

9. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

10. STROUD Worked examples and exercises are in the text Equal matrices Programme 5: Matrices Two matrices are equal if corresponding elements throughout are equal.

11. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

12. STROUD Worked examples and exercises are in the text Addition and subtraction of matrices Programme 5: Matrices Two matrices are added (or subtracted) by adding (or subtracting) corresponding elements. For example:

13. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

14. STROUD Worked examples and exercises are in the text Multiplication of matrices Scalar multiplication Multiplication if two matrices Programme 5: Matrices

15. STROUD Worked examples and exercises are in the text Multiplication of matrices Scalar multiplication Programme 5: Matrices To multiply a matrix by a single number (a scalar), each individual element of the matrix is multiplied by that number. For example: That is:

16. STROUD Worked examples and exercises are in the text Multiplication of matrices Multiplication if two matrices Programme 5: Matrices Two matrices can only be multiplied when the number of columns in the first matrix equals the number of rows in the second matrix. The ijth element of the product matrix is obtained by multiplying each element in the ith row of the first matrix by the corresponding element in the jth column of the second matrix. For example:

17. STROUD Worked examples and exercises are in the text Multiplication of matrices Multiplication if two matrices Programme 5: Matrices

18. STROUD Worked examples and exercises are in the text Multiplication of matrices Multiplication if two matrices Programme 5: Matrices

19. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

20. STROUD Worked examples and exercises are in the text Transpose of a matrix Programme 5: Matrices If a new matrix is formed by interchanging rows and columns the new matrix is called the transpose of the original matrix. For example, if:

21. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

22. STROUD Worked examples and exercises are in the text Special matrices Square matrix Diagonal matrix Unit matrix Null matrix Programme 5: Matrices

23. STROUD Worked examples and exercises are in the text Special matrices Square matrix A square matrix is of order m × m. A square matrix is symmetric if. For example: A square matrix is skew-symmetric if. For example: Programme 5: Matrices

24. STROUD Worked examples and exercises are in the text Special matrices Diagonal matrix A diagonal matrix is a square matrix with all elements zero except those on the leading diagonal. For example: Programme 5: Matrices

25. STROUD Worked examples and exercises are in the text Special matrices Unit matrix A unit matrix is a diagonal matrix with all elements on the leading diagonal being equal to unity. For example: The product of matrix A and the unit matrix is the matrix A: Programme 5: Matrices

26. STROUD Worked examples and exercises are in the text Special matrices Null matrix A null matrix is one whose elements are all zero. Notice that But that if we cannot deduce that or Programme 5: Matrices

27. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

28. STROUD Worked examples and exercises are in the text Determinant of a square matrix Singular matrix Cofactors Adjoint of a square matrix Programme 5: Matrices

29. STROUD Worked examples and exercises are in the text Determinant of a square matrix Singular matrix Programme 5: Matrices Every square matrix has its associated determinant. For example, the determinant of The determinant of a matrix is equal to the determinant of its transpose. A matrix whose determinant is zero is called a singular matrix.

30. STROUD Worked examples and exercises are in the text Determinant of a square matrix Cofactors Programme 5: Matrices Each element of a square matrix has a minor which is the value of the determinant obtained from the matrix after eliminating the ith row and jth column to which the element is common. The cofactor of element is then given as the minor of multiplied by

31. STROUD Worked examples and exercises are in the text Determinant of a square matrix Adjoint of a square matrix Programme 5: Matrices Let square matrix C be constructed from the square matrix A where the elements of C are the respective cofactors of the elements of A so that if: Then the transpose of C is called the adjoint of A, denoted by adjA.

32. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

33. STROUD Worked examples and exercises are in the text Inverse of a square matrix Programme 5: Matrices If each element of the adjoint of a square matrix A is divided by the determinant of A then the resulting matrix is called the inverse of A, denoted by A -1. Note: if det A = 0 then the inverse does not exist

34. STROUD Worked examples and exercises are in the text Inverse of a square matrix Product of a square matrix and its inverse Programme 5: Matrices The product of a square matrix and its inverse is the unit matrix:

35. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

36. STROUD Worked examples and exercises are in the text Solution set of linear equations Programme 5: Matrices The set of n simultaneous linear equations in n unknowns can be written in matrix form as:

37. STROUD Worked examples and exercises are in the text Solution set of linear equations Programme 5: Matrices Since: The solution is then:

38. STROUD Worked examples and exercises are in the text Solution set of linear equations Gaussian elimination method for solving a set of linear equations Programme 5: Matrices Given: Create the augmented matrix B, where:

39. STROUD Worked examples and exercises are in the text Solution set of linear equations Gaussian elimination method for solving a set of linear equations Programme 5: Matrices Eliminate the elements other than a 11 from the first column by subtracting a 21 /a 11 times the first row from the second row, a 31 /a 11 times the first row from the third row, etc. This gives a new matrix of the form:

40. STROUD Worked examples and exercises are in the text Solution set of linear equations Gaussian elimination method for solving a set of linear equations Programme 5: Matrices This process is repeated to eliminate the c i2 from the third and subsequent rows until a matrix of the following form is arrived at:

41. STROUD Worked examples and exercises are in the text Solution set of linear equations Gaussian elimination method for solving a set of linear equations Programme 5: Matrices From this augmented matrix we revert to the product:

42. STROUD Worked examples and exercises are in the text Solution set of linear equations Gaussian elimination method for solving a set of linear equations Programme 5: Matrices From this product the solution is derived by working backwards from the bottom starting with:

43. STROUD Worked examples and exercises are in the text Matrices – definitions Matrix notation Equal matrices Addition and subtraction of matrices Multiplication of matrices Transpose of a matrix Special matrices Determinant of a square matrix Inverse of a square matrix Solution set of linear equations Eigenvalues and eigenvectors Programme 5: Matrices

44. STROUD Worked examples and exercises are in the text Eigenvalues and eigenvectors Programme 5: Matrices Equations of the form: where A is a square matrix and is a number (scalar) have non-trivial solutions (x  0) for x called eigenvectors or characteristic vectors of A. The corresponding values of are called eigenvalues, characteristic values or latent roots of A.

45. STROUD Worked examples and exercises are in the text Eigenvalues and eigenvectors Programme 5: Matrices Expressed as a set of separate equations: That is:

46. STROUD Worked examples and exercises are in the text Eigenvalues and eigenvectors Programme 5: Matrices These can be rewritten as: That is: Which means that, for non-trivial solutions:

47. STROUD Worked examples and exercises are in the text Eigenvalues and eigenvectors Eigenvalues Programme 5: Matrices To find the eigenvalues of: solve the characteristic equation: That is: This gives eigenvalues

48. STROUD Worked examples and exercises are in the text Eigenvalues and eigenvectors Eigenvectors Programme 5: Matrices To find the eigenvectors of solve the equation For the eigenvalues = 1 and = 5

49. STROUD Worked examples and exercises are in the text Learning outcomes Define a matrix Understand what is meant by the equality of two matrices Add and subtract two matrices Multiply a matrix by a scalar and multiply two matrices together Obtain the transpose of a matrix Recognize the special types of matrix Obtain the determinant, cofactors and adjoint of a square matrix Obtain the inverse of a non-singular matrix Use matrices to solve a set of linear equations using inverse matrices Use the Gaussian elimination method to solve a set of linear equations Evaluate eigenvalues and eigenvectors Programme 5: Matrices