Mathematics Higher Level Robert Joinson

©Sumbooks 2002Higher Level49. Matrix Transformations1. By first drawing triangle A(1,1), B(3,1), C(3,4) use each of the following matrices to transformit. State the transformation each represents.⎛a)– 1 0 ⎞ ⎛b)20⎞ ⎛⎜ ⎟c)– 1 0 ⎞ ⎛⎜ ⎟d)01⎞⎜ ⎟ ⎜ ⎟⎝ 0 – 1 ⎠ ⎝ 02⎠⎝ 0 1 ⎠ ⎝ 10⎠⎛2. A transformation is given by the matrix P =– 1 1 ⎞⎜ ⎟ . Find P – 1 , the inverse of P and use it⎝ 1 1 ⎠to find the co-ordinates of the point whose image is (1,3).3. A' is the image of triangle A after it has been reflected about the y axis. A'' is the image ofthe triangle A' after it has been rotated 180° about the origin (0,0).a) If triangle A has co-ordinates (1,1), (2,1), and (1,3) draw triangle A and the images A'and A'' .b) Find the 2× 2 matrix M associated with the transformation A to A' .c) Find the 2× 2 matrix N associated with the transformation A' to A'' .d) Calculate the matrix product NM and state the single transformation which is defined bythis matrix.4. Triangle T has co-ordinates (1,1), (3,2), and (1,2). Triangle T' is the image of triangle Twhen rotated through 180° about the origin.a) What is the matrix M associated with the transformation of T to T' ?⎛Triangle T' is further transformed to T'' by the matrix N =0 – 1 ⎞⎜ ⎟⎝ 1 0 ⎠b) What transformation does matrix N perform?c) State the single transformation which is defined by the matrix product NM and writedown this matrix.5. Triangle A(1,1), B(2,2), C(1,4) is transformed onto triangle A 1 B 1 C 1 by the matrix M⎛where M =1 0 ⎞⎜ ⎟ .⎝ 0 – 1 ⎠A 1B 1C 1C 2⎛Triangle A 1 B 1 C is transformed by the matrix N where N =0 – 1 ⎞1 ⎜ ⎟⎝ – 1 0 ⎠onto the triangleA 1 B 1 C 2 .a) Find the co-ordinates of , , and and draw the three triangles.b) What single transformation will transform triangle ABC onto triangle A 1 B 1 C 2?c) By calculating the matrix W = NM show that this matrix represents the transformationfound in part ‘b’.d) Calculate W 1 , the inverse of W. Use it to find the co-ordinates of the point whose imageis (3,–5).