1. Matrices
A matrix is an array of numbers, the size of which is described by its dimensions:
An n x m matrix has n rows and m columns
Eg write down the dimensions of these matrices:

2. Matrix addition & subtraction
This can only take place between matrices of the same dimension, by simply adding or subtracting values in corresponding positions in the matrix array.
Eg if
and
, find A + B and A – B
Eg if
and
, find C + D and C – D
Now try Ex4A

3. Scalar multiplication
Multiply every number in the array by the scalar.
Eg if
, find 3A and -2A
Eg if
, find 5C and -C
Now try Ex4B

4. Requires a form of ‘cross multiplication’ and careful calculation:
Matrix multiplication
Requires a form of ‘cross multiplication’ and careful calculation: If
and
, then C1 C2 R1 R2
NB: Matrix multiplication is commutative - order matters. AB does not equal BA.
Eg if and

5. Matrix multiplication can only take place if the number of columns of the matrix on the left is equal to the number of rows of the matrix on the right.
Eg for the matrices given, which pairs can be multiplied?
Possible: AC
, BD
, CB
, DA
, DC
The product PQ will have the number of rows of P and the number of columns of Q
Eg for the possible products in the example above,
a) write down the dimensions of the products
b) Evaluate the products
Product will be 1 x 2
Product will be 2 x 1
Product will be 2 x 3
Product will be 3 x 2
Now try Ex4C

6. Matrix multiplication
NB: Matrix multiplication is associative, meaning A(BC) = (AB)C
Eg if , and

7. Inverse matrices Inverse of is where Eg Find M–1.
A matrix M is described as singular if det(M) = 0 Eg
Show that M is singular
A singular matrix has no inverse

8. WB22
, where a is a real constant.
(a) Given that a = 2, find M–1.
Inverse of is
where
(b) Find the values of a for which M is singular.
Singular

9. WB24
, where a is real.
(a) Find det A in terms of a.
(b) Show that the matrix A is non-singular for all values of a.
Singular
Has no real roots
Given that a = 0,
(c) find A–1.
Inverse of is

10. Why do inverse matrices work?

11. Using inverse matrices
Eg Ex 4G Q4a) Given that ABC = I, prove that B-1 = CA
ABC = I
Multiply on the left by A-1
BC = A-1
Multiply on the right by A
BCA = I
Multiply on the left by B-1
CA = B-1
b) Given that and , find B
Using (a),
CA = B-1
Then B is the inverse of this
Now try Ex4G, Q5, 6, 7

12. Q5a) Given that AB = C, find an expression for B
Multiply on the left by A-1
B = A-1C
b) Given that and , find B
Using (a),
B = A-1C

13. Q6a) Given that BAC = B, where B is a non-singular matrix, find an expression for A
b) When , find A
Multiply on the left by B-1
AC = I
Multiply on the right by C-1
A = C-1
Using (a),
A = C-1
7) The matrix and , Find the matrix B
B = A-1AB

14. Eg transform the shape T shown below using the matrix
Matrix transformations
Eg transform the shape T shown below using the matrix
You can express coordinates as column vectors
Use matrix multiplication to transform the shape AT T
Giving:
Original point
Image
(1,1)
(2,2)
The geometrical transformation represented by the matrix
(2,3)
(4,6)
(5,2)
(10,4)
is an enlargement about the origin, scale factor k

15. Matrix transformations
The effect of a matrix transformation can be established by considering what happens to just the coordinates (1,0) and (0,1).
The coordinates (1,0) and (0,1) are represented by the identity matrix
Applying any matrix to this will give the matrix itself
Hence the effect of a matrix can be found by taking the 1st and 2nd columns as the positions taken by the coordinates (1,0) and (0,1) respectively, and interpreting a diagram of this.
(0,1) becomes (-1,0)
Eg Given that
describe fully the geometrical transformation represented by C,
Before
After
(1,0) becomes (0,1)
Rotation 90o anticlockwise about the origin

16. Matrix transformations
Compare with the coordinates (1,0) and (0,1)
Describe fully the geometrical transformation represented by the given matrices:
Reflection in the x - axis
Rotation 180o about (0,0)
Reflection in the line y=x
Rotation 90o clockwise about the origin
Reflection in the y - axis
Reflection in the line y=-x

17. Compare with the coordinates (1,0) and (0,1)
Matrix transformations
Compare with the coordinates (1,0) and (0,1)
Describe fully the geometrical transformation represented by the given matrices:

18. Rotations by multiples of 45oEg
Describe fully the geometrical transformation represented by the matrix A.
Before
After
Rotation 135o anticlockwise about the origin
The transformation given by a power of a matrix can be understood by considering its geometric representation
Describe the transformation represented by:
Now try Ex4E
a) A2
Two 135o rotations gives a 270o rotation
b) A-1
The inverse of an anticlockwise rotation is a clockwise rotation

19. WB21
(a) Find AB.
Product will be 2 x 2
Given that
(b) describe fully the geometrical transformation represented by C,
Before
After
Reflection in y - axis
(c) write down C100.
Doing C again will return the images to their original position, as will any even power of C

20. WB23(a) Given that
(i) find A2,
(ii) describe fully the geometrical transformation represented by A2.
An enlargement about the origin, scale factor 3
(b) Given that
Before
After
describe fully the geometrical transformation represented by B.
Reflection in line
(c) Given that
where k is a constant, find the value of k for which the matrix C is singular.
Singular

21. WB28
(a) Describe fully the geometrical transformation represented by the matrix M.
Before
After
Rotation 45o anticlockwise about the origin
The transformation represented by M maps the point A with coordinates (p, q) onto the point B with coordinates (3√2, 4√2).
(b) Find the value of p and the value of q.

22. (c) Find, in its simplest surd form, the length OA, where O is the origin.
(d) Find M2.
The point B is mapped onto the point C by the transformation represented by M2.
(e) Find the coordinates of C.

23. Matrix transformations using the formulae booklet
The formulae booklet provides you with general rules:
Anticlockwise rotation through θ about 0:
Reflection in the line :
Eg to rotate 90o anticlockwise about 0
Eg to reflect in the y = x
Eg to reflect in the x-axis
Now try Ex4E, Q4-5

24. Combined transformationsEg
Find AB
A represents a rotation 90o clockwise about the origin.
B represents a reflection in the y – axis.
Hence describe fully the geometrical transformation represented by AB
Consider a triangle T
AB represents a reflection in the line y = x T BT
Eg using the same A and B above, write down: B7
A20
7 reflections in the y-axis is the same as 1
20 rotations of 90o will get back to where you started
ABT
Now try Ex4F, Q2 onwards

25. WB26 Write down the 2 × 2 matrix that represents
(a) an enlargement with centre (0, 0) and scale factor 8,
(b) a reflection in the x-axis.
Change sign of y coordinate only
(c) Hence, or otherwise, find the matrix T that represents an enlargement with centre (0, 0) and scale factor 8, followed by a reflection in the x-axis.
and
where k and c are constants.
(d) Find AB.
Given that AB represents the same transformation as T,
(e) find the value of k and the value of c.

26. Using inverse matrices in geometric situations
Eg A triangle P is transformed to a triangle Q by the matrix M
The triangle Q has vertices at the points (3, -1), (7, -9) and (24, 2). Find the coordinates of the vertices of P.
You can apply M-1 to the image to find the original points
Coordinates are
(1,-1), (3,-5), (7,-4)

27. Determining the area If then
If a shape is transformed by a matrix M, then:
Area factor =

28. WB25
, where a and b are constants.
Given that the matrix A maps the point with coordinates (4, 6) onto the point with coordinates (2, −8),
(a) find the value of a and the value of b.
A quadrilateral R has area 30 square units.
It is transformed into another quadrilateral S by the matrix A.
Using your values of a and b,
(b) find the area of quadrilateral S.
Area factor of transformation under M =
where

29. WB27
(a) Find det A.
(b) Find A–1.
Inverse of is
The triangle R is transformed to the triangle S by the matrix A.
Given that the area of triangle S is 72 square units,
(c) find the area of triangle R.
Area factor =
Area of R
The triangle S has vertices at the points (0, 4), (8, 16) and (12, 4).
(d) Find the coordinates of the vertices of R.
Coordinates (2,2), (14,10), (11,5)
Now try Ex4I

30. Using inverse matrices to solve simultaneous equations
Eg use an inverse matrix to solve the simultaneous equations
Multiply on the left by M-1
Solution
Now try Ex4J

31. Rotation 90o anticlockwise about (0,0)
Matrix algebra
Using
Compare with
Describe fully the geometrical transformation represented by B
Before
After
Rotation 90o anticlockwise about (0,0)
The triangle R is transformed to the triangle S by the matrix C.
Given that the area of triangle R is 6 square units, find the area of triangle S.
Area factor =
Area of S
The triangle S has vertices at the points (-2,2), (-10,2) and (-8,6). Find the coordinates of the vertices of R.
Coordinates
(1,1), (1,5), (3,4)