MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS
Wednesday, January 25, 2023
Form Four
MATHEMATICS
The transformation matrix transforms a vector into another vector, which can be understood geometrically in a twodimensional or a threedimensional space. The frequently used transformations are stretching, squeezing, rotation, reflection, and orthogonal projection.
MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS
IntroductionThe Unit Square
 A transformation change the shape, position or size of an object as discussed in book two.
 Pre – multiplication of any 2 x 1 column vector by a 2 x 2 matrix results in a 2 x 1 column vector
Example 1
If the vector is thought of as a position vector that is to mean that it is representing the points with coordinates (7, 1 ) to the point (17, 9).
Note;  The transformation matrix has an effect on each point of the plan. Let’s make T a transformation matrix Then T maps points (x, y) onto image points x1 ,y
Finding the Matrix of Transformation
 The objective is to find the matrix of given transformation.
Example 2
Find the matrix of transformation of triangle PQR with vertices P (1, 3) Q (3, 3) and R (2, 5).The vertices of the image of the triangles is P1(1,3) ,Q1(3,3) and R1(2,5).
Solution
Let the matrix of the transformation be
Equating the corresponding elements and solving simultaneously
a + 3b = 1
3a + 3b = 3
2a= 2
a = 1 and b = 0
c + 3d = 3
3c + 3d = 3
2c= 0
c = 0 and d = 1
Therefore the transformation matrix is
Example 3
A trapezium with vertices A (1 ,4) B(3,1 ) C (5,1 ) and D(7,4) is mapped onto a trapezium whose vertices are A1(4,1) ,B1(1 ,3) ,C1(1,5) ,D1(4 ,7).Describe the transformation and find its matrix
Solution
Let the matrix of the transformation be
Equating the corresponding elements we get;
a + 4b =  4 c + 4d = 1
3a + b = 1 3c + d = 3
Solve the equations simulteneously
3a + 12b =  12
3a + b =  1
11b = 11
hence b =1 or a = 0
3c + 12d = 3
3c + d =3
11d = 0
d = 0 c = 1
The matrix of the transformation is therefore
The transformation is positive quarter turn about the origin
Note;  Under any transformation represented by a 2 x 2 matrix, the origin is invariant, meaning it does not change its position.Therefore if the transformtion is a rotation it must be about the origin or if the transformation is reflection it must be on a mirror line which passses through the origin.
The Unit Square
 The unit square ABCD with vertices A (0,0) ,B(1,0) ,C(1,1) and D(0,1) helps us to get the transformation of a given matrix and also to identify what trasformation a given matrix represent.
Example 4
Find the images of I and J under the trasformation whose matrix is;
Solution
NOTE;  The images of I and J under transformation represented by any 2 x 2 matrix i.e., are I1(a ,c) and J1(b ,d)
Example 5
Find the matrix of reflection in the line y = 0 or x axis.
Solution
Using a unit square the image of B is (1 , 0) and D is (0 , 1 ) .Therefore , the matrix of the transformation is
 Example 6
Show on a diagram the unit square and it image under the transformation represented by the matrix
Solution
Using a unit square, the image of I is ( 1 ,0 ), the image of J is ( 4 , 1 ),the image of O is ( 0,0) and that of K is
Therefore ,K1,the image of K is ( 5 ,1)
Successive Transformations
 The process of performing two or more transformations in order is called successive transformation e.g. performing transformation H followed by transformation Y is written as follows YH or if A, B and C are transformations; then ABC means perform C first,then B and finally A , in that order.
 The matrices listed below all perform different rotations/reflections:
 This transformation matrix is the identity matrix. When multiplying by this matrix, the point matrix is unaffected and the new matrix is exactly the same as the point matrix
 This transformation matrix creates a reflection in the xaxis. When multiplying by this matrix, the x coordinate remains unchanged, but the y coordinate changes sign.
 This transformation matrix creates a reflection in the yaxis. When multiplying by this matrix, the y coordinate remains unchanged, but the x coordinate changes sign.
 This transformation matrix creates a rotation of 1 80 degrees. When multiplying by this matrix, the point matrix is rotated 1 80 degrees around (0, 0). This changes the sign of both the x and y coordinates.
 This transformation matrix creates a reflection in the line y=x. When multiplying by this matrix, the x coordinate becomes the y coordinate and the yordinate becomes the x coordinate
 This transformation matrix rotates the point matrix 90 degrees clockwise. When multiplying by this matrix, the point matrix is rotated 90 degrees clockwise around (0, 0).
 This transformation matrix rotates the point matrix 90 degrees anticlockwise. When multiplying by this matrix, the point matrix is rotated 90 degrees anticlockwise around (0, 0).
 This transformation matrix creates a reflection in the line y=x. When multiplying by this matrix, the point matrix is reflected in the line y=x changing the signs of both coordinates and swapping their values.
Inverse Matrix Transformation
 A transformation matrix that maps an image back to the object is called an inverse of matrix.
Note;  If A is a transformation which maps an object T onto an image T1,then a transformation that can map T1 back to T is called the inverse of the transformation A , written as image A1.
 If R is a positive quarter turn about the origin the matrix for R is and the matrix for R1 is hence R1R = RR1 = 1
Example 7
T is a triangle with vertices A (2, 4), B (1 , 2) and C (4, 2).S is a transformation represented by the matrix Draw T and its image T1 under the transformation S
 Find the matrix of the inverse of the transformation S
Solution  Using transformation matrix S =
 Let the inverse of the transformation matrix be(ac db). This can be done in the following ways
 S1S = 1
Therefore
Equating corresponding elements and solving simultaneously;
a = 1 ,b = 2 , c = 0 and d = 2
Therefore

Area Scale Factor and Determinant of Matrix
 The ratio of area of image to area object is the area scale factor (A.S.F)
Area scale factor = area of image
area of object  Area scale factor is numerically equal to the determinant. If the determinant is negative you simply ignore the negative sign.
Example 8
Area of the object is 4 cm and that of image is 36 cm find the area scale factor.
Solution
36 = 9
4
If it has a matrix of the determinant is 9  0 = 9 hence equal to A.S.F
Shear and Stretch
Shear
 The transformation that maps an object (in orange) to its image (in blue) is called a shear
 The object has same base and equal heights. Therefore, their areas are equal. Under any shear, area is always invariant (fixed)
 A shear is fully described by giving;
 The invariant line
 A point not on the invariant line, and its image.
Example 9
A shear X axis invariant
Example 10
A shear Y axis invariant
Note;  Shear with x axis invariant is represented by a matrix of the form under this trasnsformation, J (0, 1 ) is mapped onto J1(k,1).
 Likewise a shear with y – axis invariant is represented by a matrix of the form . Under this transformation, I (0,1 ) is mapped onto I1(1,k).
Stretch
 A stretch is a transformation which enlarges all distance in a particular direction by a constant factor. A stretch is described fully by giving;
 The scale factor
 The invariant line
 Note;
 If K is greater than 1 , then this really is a stretch.
 If k is less than one 1 , it is a squish but we still call it a stretch
 If k = 1 , then this transformation is really the identity i.e. it has no effect.
Example 11
Using a unit square, find the matrix of the stretch with y axis invariant ad scale factor 3
Solution
The image of I is I1(1, 0) and the image of J is (0,1) therefore the matrix of the stretch is  Note;
The matrix of the stretch with the yaxis invariant and scale factor k is and the matrix of a stretch with x – axis invariant and scale factor k is
Isometric and Nonisometric Transformation
 Isometric transformations are those in which the object and the image have the same shape and size (congruent) e.g. rotation, reflection and translation
 Non isometric transformations are those in which the object and the image are not congruent e.g., shear stretch and enlargement
Past KCSE Questions on the Topic
 Matrix p is given by . Find P1
 A triangle is formed by the coordinates A (2, 1 ) B (4, 1 ) and C (1 , 6). It is rotated clockwise through 90o about the origin. Find the coordinates of this image.
 On the grid provided, A (1 , 2) B (7, 2) C (4, 4) D (3, 4) is a trapezium
 ABCD is mapped onto A’B’C’D’ by a positive quarter turn. Draw the image A’B’C’D on the grid
 A transformation matrix maps A’B’C’D onto A”B” C”D” Find the coordinates of A”B”C”D”
 A triangle T whose vertices are A (2, 3) B (5, 3) and C (4, 1 ) is mapped onto triangle T1 whose vertices are A1 (4, 3) B1 (1 , 3) and C1 (x, y) by a Transformation M =
 Find the:
 Matrix M of the transformation
 Coordinates of C1
 Triangle T2 is the image of triangle T1 under a reflection in the line y = x. Find a single matrix that maps T and T2
 Find the:
 Triangles ABC is such that A is (2, 0), B (2, 4), C (4, 4) and A”B”C” is such that A” is (0, 2), B” (4 ,–10) and C “is (4, 12) are drawn on the Cartesian plane.
Triangle ABC is mapped onto A”B”C” by two successive transformations
R=Followed by Find R
 Using the same scale and axes, draw triangles A’B’C’, the image of triangle ABC under transformation R. Describe fully, the transformation represented by matrix R
 Triangle ABC is shown on the coordinate’s plane below
 Given that A (6, 5) is mapped onto A (6,4) by a shear with y axis invariant
 Draw triangle A’B’C’, the image of triangle ABC under the shear
 Determine the matrix representing this shear
 Draw triangle A’B’C’, the image of triangle ABC under the shear
 Triangle A B C is mapped on to A” B” C” by a transformation defined by the matrix
 Draw triangle A” B” C”
 Describe fully a single transformation that maps ABC onto A”B” C”
 Given that A (6, 5) is mapped onto A (6,4) by a shear with y axis invariant
 Determine the inverse T1of the matrix T =
Hence find the coordinates to the point at which the two lines x + 2y = 7 and x  y =1  Given that A = B =
Find the value of x if A 2x = 2B
 3x – 2A = 3B
 2A  3B = 2x
 A 2x = 2B
 The transformation R given by the matrix
 Determine the matrix A giving a, b, c and d as fractions
 Given that A represents a rotation through the origin, determine the angle of rotation.
 S is a rotation through 180 about the point (2, 3). Determine the image of (1 , 0) under S followed by R
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