PROGRAMME 4 DETERMINANTS

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

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

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.

Determinants Therefore: That is:

Determinants The three determinants: can be obtained from the two equations as follows:

Determinants The equations: can then be written as:

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

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

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:

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:

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:

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Simultaneous equations in three unknowns The equations: have solution: More easily remembered as:

Simultaneous equations in three unknowns where: from the equations:

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of 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:

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Determinants Determinants of third order Simultaneous equations in three unknowns Consistency of a set of equations Properties of determinants

Properties of determinants The value of a determinant remains unchanged if rows are changed to columns and columns changed to rows:

Properties of determinants 2. If two rows (or columns) are interchanged, the sign of the determinant is changed:

Properties of determinants 3. If two rows (or columns) are identical, the value of the determinant is zero:

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:

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:

Learning outcomes Expand a 2 × 2 determinant Solve pairs of simultaneous equations in two variables using 2 × 2 determinants Expand a 3 × 3 determinant Solve three simultaneous equations in three variables using 3 × 3 determinants Determine the consistency of sets of simultaneous linear equations Use the properties of determinants to solve equations written in determinant form