1. PROGRAMME 4
DETERMINANTS

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

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

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

5. 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.

6. Determinants
Therefore:
That is:

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

8. Determinants
The equations:
can then be written as:

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

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

11. 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

12. 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:

13. 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:

14. 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:

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

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

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

18. Simultaneous equations in three unknowns
where:
from the equations:

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

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

21. 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:

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

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

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

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

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

27. 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:

28. 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:

29. 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