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:

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

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

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

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

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

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

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

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

Why do inverse matrices work?

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

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

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

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

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

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

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:

Rotations by multiples of 45o Eg 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

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

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

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.

(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.

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

Combined transformations Eg 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

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.

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)

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

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

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

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

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)