Programme 4 Determinants PROGRAMME 4 DETERMINANTS STROUD Worked

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Programme 4: Determinants PROGRAMME 4 DETERMINANTS STROUD Worked examples and exercises are in the

Programme 4: Determinants PROGRAMME 4 DETERMINANTS STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants Solving the two simultaneous equations: results in: which has a solution

Programme 4: Determinants Solving the two simultaneous equations: results in: which has a solution provided STROUD Worked examples and exercises are in the text

Programme 4: Determinants There is a shorthand notation for . It is: The symbol:

Programme 4: Determinants There is a shorthand notation for . It is: The symbol: (evaluated by cross multiplication as ) Is called a second-order determinant; second-order because it has two rows and two columns. STROUD Worked examples and exercises are in the text

Programme 4: Determinants Therefore: That is: STROUD Worked examples and exercises are in the

Programme 4: Determinants Therefore: That is: STROUD Worked examples and exercises are in the text

Programme 4: Determinants The three determinants: from the two equations STROUD can be obtained

Programme 4: Determinants The three determinants: from the two equations STROUD can be obtained as follows: Worked examples and exercises are in the text

Programme 4: Determinants The equations: can then be written as: STROUD Worked examples and

Programme 4: Determinants The equations: can then be written as: STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Minors Evaluation of a determinant of third-order about

Programme 4: Determinants of third order Minors Evaluation of a determinant of third-order about the first row Evaluation of a determinant about any row or column STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Minors A third-order determinant has three rows and

Programme 4: Determinants of third order Minors A third-order determinant has three rows and three columns. Each element of the determinant has an associated minor – a second order determinant obtained by eliminating the row and column to which it is common. For example: STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Evaluation of a third-order determinant about the first

Programme 4: Determinants of third order Evaluation of a third-order determinant about the first row To expand a third-order determinant about the first row we multiply each element of the row by its minor and add and subtract the products as follows: STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Evaluation of a determinant about any row or

Programme 4: Determinants of third order Evaluation of a determinant about any row or column To expand a determinant about any row or column we multiply each element of the row or column by its minor and add and subtract the products according to the pattern: STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants Simultaneous equations in three unknowns The equations: have solution: More easily

Programme 4: Determinants Simultaneous equations in three unknowns The equations: have solution: More easily remembered as: STROUD Worked examples and exercises are in the text

Programme 4: Determinants Simultaneous equations in three unknowns where: from the equations: STROUD Worked

Programme 4: Determinants Simultaneous equations in three unknowns where: from the equations: STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants Consistency of a set of equations The three equations in two

Programme 4: Determinants Consistency of a set of equations The three equations in two unknowns are consistent if they possess a common solution. That is: have a common solution and are, therefore, consistent if: STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a

Programme 4: Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants STROUD Worked examples and exercises are in the text

Programme 4: Determinants Properties of determinants 1. The value of a determinant remains unchanged

Programme 4: Determinants Properties of determinants 1. The value of a determinant remains unchanged if rows are changed to columns and columns changed to rows: STROUD Worked examples and exercises are in the text

Programme 4: Determinants Properties of determinants 2. If two rows (or columns) are interchanged,

Programme 4: Determinants Properties of determinants 2. If two rows (or columns) are interchanged, the sign of the determinant is changed: STROUD Worked examples and exercises are in the text

Programme 4: Determinants Properties of determinants 3. If two rows (or columns) are identical,

Programme 4: Determinants Properties of determinants 3. If two rows (or columns) are identical, the value of the determinant is zero: STROUD Worked examples and exercises are in the text

Programme 4: Determinants Properties of determinants 4. If the elements of any one row

Programme 4: Determinants Properties of determinants 4. If the elements of any one row (or column) are all multiplied by a common factor, the determinant is multiplied by that factor: STROUD Worked examples and exercises are in the text

Programme 4: Determinants Properties of determinants 5. If the elements of any one row

Programme 4: Determinants Properties of determinants 5. If the elements of any one row (or column) are increased by equal multiples of the corresponding elements of any other row (or column), the value of the determinant is unchanged: STROUD Worked examples and exercises are in the text

Programme 4: Determinants Learning outcomes üExpand a 2 × 2 determinant üSolve pairs of

Programme 4: Determinants Learning outcomes üExpand a 2 × 2 determinant üSolve pairs of simultaneous equations in two variables using 2 × 2 determinants üExpand a 3 × 3 determinant üSolve three simultaneous equations in three variables using 3 × 3 determinants üDetermine the consistency of sets of simultaneous linear equations üUse the properties of determinants to solve equations written in determinant form STROUD Worked examples and exercises are in the text

Programme 5 Matrices PROGRAMME 5 MATRICES STROUD WorkedProgramme 5 Matrices PROGRAMME 5 MATRICES STROUD Worked
Programme 7 Differentiation PROGRAMME 7 DIFFERENTIATION STROUD WorkedProgramme 7 Differentiation PROGRAMME 7 DIFFERENTIATION STROUD Worked
Programme 6 Vectors PROGRAMME 6 VECTORS STROUD WorkedProgramme 6 Vectors PROGRAMME 6 VECTORS STROUD Worked
Stroud District Local Plan Briefing on Stroud DistrictStroud District Local Plan Briefing on Stroud District
PROGRAMME 14 SERIES 2 STROUD Worked examples andPROGRAMME 14 SERIES 2 STROUD Worked examples and
PROGRAMME F 10 FUNCTIONS STROUD Worked examples andPROGRAMME F 10 FUNCTIONS STROUD Worked examples and
PROGRAMME 12 CURVES AND CURVE FITTING STROUD WorkedPROGRAMME 12 CURVES AND CURVE FITTING STROUD Worked
PROGRAMME 1 COMPLEX NUMBERS 1 STROUD Worked examplesPROGRAMME 1 COMPLEX NUMBERS 1 STROUD Worked examples
PROGRAMME 27 STATISTICS STROUD Worked examples and exercisesPROGRAMME 27 STATISTICS STROUD Worked examples and exercises
PROGRAMME 3 HYPERBOLIC FUNCTIONS STROUD Worked examples andPROGRAMME 3 HYPERBOLIC FUNCTIONS STROUD Worked examples and
PROGRAMME 8 DIFFERENTIATION APPLICATIONS 1 STROUD Worked examplesPROGRAMME 8 DIFFERENTIATION APPLICATIONS 1 STROUD Worked examples
PROGRAMME 18 INTEGRATION APPLICATIONS 1 STROUD Worked examplesPROGRAMME 18 INTEGRATION APPLICATIONS 1 STROUD Worked examples
PROGRAMME F 8 TRIGONOMETRY STROUD Worked examples andPROGRAMME F 8 TRIGONOMETRY STROUD Worked examples and
PROGRAMME 27 STATISTICS STROUD Worked examples and exercisesPROGRAMME 27 STATISTICS STROUD Worked examples and exercises
PROGRAMME F 2 INTRODUCTION TO ALGEBRA STROUD WorkedPROGRAMME F 2 INTRODUCTION TO ALGEBRA STROUD Worked
PROGRAMME F 2 INTRODUCTION TO ALGEBRA STROUD WorkedPROGRAMME F 2 INTRODUCTION TO ALGEBRA STROUD Worked
PROGRAMME F 10 FUNCTIONS STROUD Worked examples andPROGRAMME F 10 FUNCTIONS STROUD Worked examples and
PROGRAMME F 11 DIFFERENTIATION STROUD Worked examples andPROGRAMME F 11 DIFFERENTIATION STROUD Worked examples and
PROGRAMME F 11 DIFFERENTIATION slope finding STROUD WorkedPROGRAMME F 11 DIFFERENTIATION slope finding STROUD Worked
PROGRAMME F 7 PARTIAL FRACTIONS STROUD Worked examplesPROGRAMME F 7 PARTIAL FRACTIONS STROUD Worked examples
PROGRAMME 23 MULTIPLE INTEGRALS STROUD Worked examples andPROGRAMME 23 MULTIPLE INTEGRALS STROUD Worked examples and
PROGRAMME 13 SERIES 1 STROUD Worked examples andPROGRAMME 13 SERIES 1 STROUD Worked examples and
PROGRAMME 8 DIFFERENTIATION APPLICATIONS STROUD Worked examples andPROGRAMME 8 DIFFERENTIATION APPLICATIONS STROUD Worked examples and
PROGRAMME F 4 GRAPHS STROUD Worked examples andPROGRAMME F 4 GRAPHS STROUD Worked examples and