Determinants

Solving simultaneous equations can be done quite easily by elimination.
However, what happens if substitution is investigated ?

 

Solving the equations

ax +by = e
cx + dy = f

1

2

Note, the numerator is  found by cross multiplying
and subtracting using the y co-efficients.

             bf                     de

3

The denominator is found by reverse cross multiplying
and subtracting the coefficients of x and y.

             bc                    ad

4

And, for y

5

6

Note, the numerator is  found by cross multiplying
and subtracting using the x co-efficients.

             af                   ce

7
The denominator is found by cross multiplying
and subtracting the coefficients of x and y.

             ad                     bc

8

This is the vedic way of solving simultaneous equations.
It will only work when ad – bc ≠ 0

 

Example, solve the following system of equations

2x +3y = 7
3x + 4y = 6

     9        10  

 

Cramer's Rule

 

ram1

cramx cramy

Example, solve the following system of equations

2x +3y = 7
3x + 4y = 6

jk

kl

 

 

 

Click for a spreadsheet to calculate a 2x2 system.

Determinants

 

The system of equations
ax + by = e
cx + dy = f

can be written in matrix form as

Ax = b

Where

11

 

Notice that the denominator from above,
ad – bc  , is found by cross multiplying and subtracting
the elements in A.

This is known as the determinant of A.
It is written det(A) or 12

13

 If det(A) = 0, the system has no solution.

Examples

Does the follow system of equations
have a solution ?

3x + 2y = 7
9x + 6y = 6

14

No solution, since determinant = 0

Determinant of a 3x3 matrix

15 16

Each element in A is associated with its minor

17

18

19

To find the determinant,

       Place signs
       20

 

21

Example

Find the determinant of

22

23

24

 

Note:

25

 

Cofactors

Cofactors are minors with their place sign.

Example

Find the cofactors of

27 which has 27

 

First row

28

29

30

Second row

31

32

33

Third row

34

35

36

Combining these with the will give a new matrix C,
made up of  the cofactors of A.

so 38

 

This is written Adj (A)

It is the transpose of C
so

Adj(A) = CT

 

Example

Find Adj (A) for the matrix

37

 

From the work above,

39

Transposed,

40

 

Inverse of a square matrix

40

 

Example

Find the inverse of the matrix

27

 

From the work above,

 

42

 

Note,

43

 

Product of a square matrix and its inverse

    44

Example, using the matrix from above

45

46

This means that Gaussian elimination can be used to find the inverse:
Write down the matrix to be inverted with the Identity matrix next to it.
Perform ERO’s to reduce the left hand matrix to I,
  simultaneously performing the same ERO’s on the Identity matrix.
The result is the inverse of the given matrix.

Example – using the question given above.

47

48

49

50

52

53

54

56

© Alexander Forrest