Representing transformations by matrices
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Teaching Further Mathematics FP1 Representing transformations by matrices Give each pair of students a copy of the sheet below enlarged onto A3. Students have to multiply the matrix by the position vector of each vertex of the triangle, plot the new position of the triangle and state the transformation that the matrix represents. This enables students to find out which transformation each matrix represents. Discussion points: • ⎛ −1 0 ⎞⎛ x ⎞ ⎛ − x ⎞ Compare ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ 0 1 ⎠⎝ y ⎠ ⎝ y ⎠ Which transformation is this? ⎛1⎞ ⎛0⎞ ⎛0⎞ What happen to the point with position vector ⎜ ⎟ ?... ⎜ ⎟ ?... ⎜ ⎟ ? ⎝0⎠ ⎝1⎠ ⎝0⎠ ⎛0⎞ What happen to the point with position vector ⎜ ⎟ ? Is this always the case? ⎝0⎠ ⎛ −1 0 ⎞ ⎛ x ⎞ ⎛ −x ⎞ ⎟ is a reflection in the line x = 0 . This could be expressed ⎜ ⎟ → ⎜ ⎟ ⎜ ⎝ 0 1⎠ ⎝ y⎠ ⎝ y ⎠ • • ⎛ x ⎞ ⎛ −x ⎞ Which transformation is ⎜ ⎟ → ⎜ ⎟ . What is the corresponding matrix? ⎝ y⎠ ⎝−y⎠ How would you write the rotation of 180D about the origin transformation in the form ⎛ x⎞ ⎛ ⎞ ⎜ ⎟ → ⎜ ⎟ ? What is the corresponding matrix transformation? ⎝ y⎠ ⎝ ⎠ © Susan Wall, MEI 2006 Page 1 of 2 Teaching Further Mathematics FP1 Transformation: Transformation: Transformation: Transformation: Transformation: ⎛1 0 ⎞ ⎜⎜ ⎟⎟ ⎝ 0 1⎠ Transformation: ⎛ 0 − 1⎞ ⎜⎜ ⎟⎟ − 1 0 ⎝ ⎠ ⎛ − 1 0⎞ ⎜⎜ ⎟⎟ 0 1 ⎝ ⎠ ⎛2 ⎜⎜ ⎝0 0⎞ ⎟ 1 ⎟⎠ ⎛0 1 ⎞ ⎜⎜ ⎟⎟ ⎝ − 1 0⎠ ⎛ 0 1⎞ ⎜⎜ ⎟⎟ ⎝1 0 ⎠ Transformation: (0, 1) ⎛ 2 0⎞ ⎜⎜ ⎟⎟ 0 2 ⎝ ⎠ Transformation: ⎛ − 1 0⎞ ⎜⎜ ⎟⎟ 0 − 1 ⎝ ⎠ (2, 0) ⎛1 0 ⎞ ⎜⎜ ⎟⎟ ⎝0 2⎠ ⎛ 0 − 1⎞ ⎜⎜ ⎟⎟ ⎝1 0 ⎠ ⎛1 0 ⎞ ⎜⎜ ⎟⎟ 0 − 1 ⎝ ⎠ Transformation: Transformation: Transformation: © Susan Wall, MEI 2006 Page 2 of 2 More Matrix Transformations Fill in the matrices for the following transformations Transformation Reflection in the x axis Matrix Transformation Rotation by 45° clockwise about the origin Reflection in the y axis ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ N ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ O ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ P Enlargement with scale factor 2 Q Rotation by 180° about the origin Rotation by 45° anticlockwise about the origin Rotation by 90° anticlockwise about the origin Rotation by 135° clockwise about the origin Reflection in the line ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ R ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ S ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ T y=x ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ U I Rotation by 360° about the origin ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ V J Stretch with a scale factor 4 parallel to x axis Reflection in the line ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ W ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ Y ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ Z A B C D Rotation by 30° clockwise about the origin Stretch with scale factor 5 parallel to y axis Reflection in the line y= E F G H K 1 x 3 y = − 3x L M Rotation by 60° clockwise about the origin Rotation by 120° anticlockwise about the origin Matrix ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ Reflection in the line ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ Rotation by 120° clockwise about the origin Rotation by 30° anticlockwise about the origin Rotation by 150° clockwise about the origin Rotation by 90° clockwise about the origin Reflection in the line ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ y = −x y = 3x X Rotation by 150° anticlockwise about the origin Enlargement with a scale factor 3 Reflection in the line 1 y=− x 3 Find the smallest positive value of n in each of the following cases: 1. 6. Ln = I Hn = I 2. 7. Mn = I Xn = I 3. 8. Vn = I Qn = I 4. 9. An = I On = I 5. 10. Gn = I Tn = I Solutions to More Matrix Transformations Transformation A Reflection in the x axis B Rotation by 30° clockwise about the origin C Stretch with scale factor 5 parallel to y axis D Reflection in the line 1 y= x 3 E Rotation by 45° anticlockwise about the origin F Rotation by 90° anticlockwise about the origin G Rotation by 135° clockwise about the origin Matrix Transformation ⎛1 0 ⎞ ⎜ 0 −1⎟ ⎝ ⎠ Rotation by 45° clockwise about the origin ⎛ 3 1 ⎞ ⎜ ⎟ 2 ⎟ ⎜ 2 ⎜ 1 3⎟ ⎜− ⎟ ⎝ 2 2 ⎠ ⎛1 0⎞ ⎜ ⎟ ⎝0 5⎠ ⎛ 1 3⎞ ⎜ ⎟ 2 ⎟ ⎜ 2 ⎜ 3 1⎟ − ⎟ ⎜ ⎝ 2 2⎠ 1 ⎞ ⎛ 1 − ⎜ ⎟ 2 2⎟ ⎜ 1 ⎟ ⎜ 1 ⎜ ⎟ 2 ⎠ ⎝ 2 O Reflection in the y axis P Enlargement with scale factor 2 Q Rotation by 180° about the origin ⎛ −1 0 ⎞ ⎜ 0 −1 ⎟ ⎝ ⎠ R Reflection in the line ⎛ 0 −1 ⎞ ⎜ −1 0 ⎟ ⎝ ⎠ ⎛ 0 −1⎞ ⎜1 0 ⎟ ⎝ ⎠ S Rotation by 120° clockwise about the origin T Rotation by 30° anticlockwise about the origin ⎛ 1 ⎜ − ⎜ 2 ⎜ 3 ⎜− ⎝ 2 ⎛ 3 ⎜ ⎜ 2 ⎜ 1 ⎜ ⎝ 2 ⎛ 3 ⎜− 2 ⎜ ⎜ 1 ⎜ − ⎝ 2 ⎛ −1 ⎜ ⎝0 3⎞ ⎟ 2 ⎟ 1⎟ − ⎟ 2⎠ 1⎞ − ⎟ 2⎟ 3⎟ ⎟ 2 ⎠ 1 ⎞ ⎟ 2 ⎟ 3⎟ − ⎟ 2 ⎠ 0⎞ ⎟ 1⎠ ⎛ 1 ⎜− ⎜ 2 ⎜ 3 ⎜ ⎝ 2 3⎞ ⎟ 2 ⎟ 1 ⎟ ⎟ 2 ⎠ ⎛ ⎜− ⎜ ⎜ ⎜− ⎝ 1 2 1 2 1 ⎞ ⎟ 2 ⎟ 1 ⎟ − ⎟ 2⎠ y = −x H Reflection in the line ⎛0 1⎞ ⎜1 0⎟ ⎝ ⎠ U Rotation by 150° clockwise about the origin I Rotation by 360° about the origin Stretch with a scale factor 4 parallel to x axis ⎛1 0⎞ ⎜ ⎟ ⎝0 1⎠ V Rotation by 90° clockwise about the origin ⎛ 4 0⎞ ⎜0 1⎟ ⎝ ⎠ W Reflection in the line ⎛ 1 3⎞ − ⎜ − ⎟ 2 ⎟ ⎜ 2 ⎜ 3 1 ⎟ ⎜− ⎟ ⎝ 2 2 ⎠ ⎛ 1 3⎞ ⎜ ⎟ 2 ⎟ ⎜ 2 ⎜ 3 1 ⎟ ⎜− ⎟ ⎝ 2 2 ⎠ ⎛ 1 3⎞ − ⎜− ⎟ 2 ⎟ ⎜ 2 ⎜ 3 1 ⎟ − ⎟ ⎜ ⎝ 2 2 ⎠ X Rotation by 150° anticlockwise about the origin Y Enlargement with a scale factor 3 Z Reflection in the line J y=x K Reflection in the line y = − 3x L Rotation by 60° clockwise about the origin M Rotation by 120° anticlockwise about the origin 1. 6 2. 3 3. 4 Matrix 1 ⎞ ⎛ 1 ⎜ ⎟ 2⎟ ⎜ 2 1 ⎟ ⎜ 1 ⎜− ⎟ 2 2⎠ ⎝ N 4. 2 5. 8 ⎛ −1 0 ⎞ ⎜ 0 1⎟ ⎝ ⎠ ⎛ 2 0⎞ ⎜ ⎟ ⎝0 2⎠ y = 3x y=− 6. 2 1 x 3 7. 12 8. 2 ⎛ 3 ⎜− 2 ⎜ ⎜ 1 ⎜ ⎝ 2 1 ⎞ ⎟ 2 ⎟ 3⎟ − ⎟ 2 ⎠ − ⎛ 3 0⎞ ⎜ 0 3⎟ ⎝ ⎠ ⎛ 1 ⎜ ⎜ 2 ⎜ 3 ⎜− ⎝ 2 9. 2 3⎞ ⎟ 2 ⎟ −1 ⎟ ⎟ 2 ⎠ − 10. 12 Teaching Further Mathematics FP1 Geometrical properties of matrix transformations: ⎛1 ⎜ ⎝0 ⎛0 ⎜ ⎝0 ⎛0 ⎜ ⎝1 ⎛0 ⎜ ⎝0 ⎛1 ⎜ ⎝0 ⎛1 ⎜ ⎝1 ⎛1 ⎜ ⎝0 ⎛0 ⎜ ⎝1 0⎞⎛ x ⎞ ⎛ x⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0⎠⎝ y ⎠ ⎝ 0⎠ Projects a point down onto the x-axis 1 ⎞⎛ x ⎞ ⎛ y ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0 ⎠⎝ y ⎠ ⎝ 0 ⎠ 0⎞⎛ x ⎞ ⎛ 0⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0⎠⎝ y ⎠ ⎝ x⎠ 0 ⎞⎛ x ⎞ ⎛ 0 ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 1 ⎠⎝ y ⎠ ⎝ y ⎠ Projects a point across onto the y-axis 1⎞⎛ x ⎞ ⎛ x + y ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0⎠⎝ y ⎠ ⎝ 0 ⎠ 0⎞⎛ x ⎞ ⎛ x⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0⎠⎝ y ⎠ ⎝ x⎠ Projects a point up/down onto the line y=x 0 ⎞⎛ x ⎞ ⎛ x ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 1 ⎠⎝ y ⎠ ⎝ y ⎠ Leaves the point unchanged 1 ⎞⎛ x ⎞ ⎛ y ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0 ⎠⎝ y ⎠ ⎝ x ⎠ Reflects in the line y=x ⎛ 0 1⎞ ⎛ x ⎞ ⎛ y ⎞ ⎜ ⎟⎜ ⎟ → ⎜ ⎟ ⎝ 0 1⎠ ⎝ y ⎠ ⎝ y ⎠ ⎛0 0⎞⎛ x ⎞ ⎛ 0 ⎞ ⎜ ⎟⎜ ⎟ → ⎜ ⎟ ⎝1 1⎠⎝ y ⎠ ⎝ x + y ⎠ ⎛1 ⎜ ⎝1 ⎛1 ⎜ ⎝0 1⎞⎛ x ⎞ ⎛ x + y ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 0⎠⎝ y ⎠ ⎝ x ⎠ 1⎞ ⎛ x ⎞ ⎛ x + y ⎞ ⎟⎜ ⎟ → ⎜ ⎟ 1⎠ ⎝ y ⎠ ⎝ y ⎠ See example below ⎛1 0 ⎞ ⎛ x ⎞ ⎛ x ⎞ ⎜ ⎟⎜ ⎟ → ⎜ ⎟ ⎝1 1 ⎠ ⎝ y ⎠ ⎝ x + y ⎠ ⎛ 0 1⎞ ⎛ x ⎞ ⎛ y ⎞ ⎟ ⎜ ⎟⎜ ⎟ → ⎜ ⎝ 1 1⎠ ⎝ y ⎠ ⎝ x + y ⎠ Notice how the line y = − x collapses to the ⎛ 1 1⎞ ⎛ x ⎞ ⎛ x + y ⎞ ⎜ ⎟⎜ ⎟ → ⎜ ⎟ origin ⎝ 1 1⎠ ⎝ y ⎠ ⎝ x + y ⎠ Investigate entries of –1,1 and 0 ⎛ −1 0 ⎞ ⎛ x ⎞ ⎛ − x ⎞ A rotation through 180º about the origin. ⎜ ⎟⎜ ⎟ → ⎜ ⎟ ⎝ 0 −1 ⎠ ⎝ y ⎠ ⎝ − y ⎠ © MEI 2006 Page 1 of 1 Teaching Further Mathematics FP1 Matrices Using the matrices given on the sheet below, ask students which they can combine and in what order, to make the following transformations: Rotation of 180º Identity Reflection in the x-axis Reflection in the line y = − x Ask them if they can make any of them in more than one way. They should justify their answers using matrix multiplication. Discussion points: • • ⎛1 0 ⎞ What is the transformation corresponding to ⎜ ⎟ ? How do you know? ⎝ 0 −1 ⎠ ⎛ 0 1⎞ ⎛ 0 −1 ⎞ ⎛0 1⎞ How about ⎜ ⎟? ⎟ ?....and ⎜ ⎟ ? and ⎜ ⎝ −1 0 ⎠ ⎝1 0 ⎠ ⎝1 0⎠ ⎛1 Example: ⎜ ⎝0 ⎛1 Method 1. ⎜ ⎝0 ⎛1 Method 2. ⎜ ⎝0 0 ⎞⎛0 ⎟⎜ −1 ⎠ ⎝ 1 0 ⎞⎛0 ⎟⎜ −1 ⎠ ⎝ 1 0 ⎞⎛0 ⎟⎜ −1 ⎠ ⎝ 1 −1 ⎞ ⎟ 0 ⎠ −1 ⎞ ⎛ x ⎞ ⎛ 0 −1 ⎞ ⎛ x ⎞ ⎛ − y ⎞ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟ = ⎜ ⎟ 0 ⎠ ⎝ y ⎠ ⎝ −1 0 ⎠ ⎝ y ⎠ ⎝ − x ⎠ −1 ⎞ ⎛ x ⎞ ⎛ 1 0 ⎞ ⎛ − y ⎞ ⎛ − y ⎞ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟=⎜ ⎟ 0 ⎠ ⎝ y ⎠ ⎝ 0 −1 ⎠ ⎝ x ⎠ ⎝ − x ⎠ • In Method 1 above, how does this show that the transformation is a reflection in the line y = −x ? • What is the relevance of the intermediate step in Method 2? • • ⎛1 0 ⎞ ⎛ 0 −1 ⎞ What are the individual transformations ⎜ ⎟ and ⎜ ⎟ ? In terms of what these ⎝ 0 −1 ⎠ ⎝1 0 ⎠ individually do, convince me what their combined effect is. How else might you convince yourself of the nature of the resulting transformation. (You might ⎛1⎞ ⎛0⎞ suggest they think what happens to ⎜ ⎟ and ⎜ ⎟ under the combined transformation.) ⎝0⎠ ⎝1⎠ © Susan Wall, MEI 2006 Page 1 of 2 Teaching Further Mathematics FP1 ⎛1 ⎜ ⎝0 ⎛1 ⎜ ⎝0 0 ⎞ ⎛ 0 ⎟ ⎜ −1 ⎠ ⎝ −1 0 ⎞ ⎛ 0 ⎟ ⎜ −1 ⎠ ⎝ −1 1⎞ ⎛0 ⎟ ⎜ 0⎠ ⎝1 1⎞ ⎛0 ⎟ ⎜ 0⎠ ⎝1 −1 ⎞ ⎛ 0 ⎟ ⎜ 0 ⎠ ⎝1 −1 ⎞ ⎛ 0 ⎟ ⎜ 0 ⎠ ⎝1 1⎞ ⎟ 0⎠ ⎛1 ⎜ ⎝0 ⎛1 ⎜ ⎝0 0 ⎞ ⎛ 0 ⎟ ⎜ −1 ⎠ ⎝ −1 0 ⎞ ⎛ 0 ⎟ ⎜ −1 ⎠ ⎝ −1 1⎞ ⎛0 ⎟ ⎜ 0⎠ ⎝1 1⎞ ⎛0 ⎟ ⎜ 0⎠ ⎝1 −1 ⎞ ⎛ 0 ⎟ ⎜ 0 ⎠ ⎝1 −1 ⎞ ⎛ 0 ⎟ ⎜ 0 ⎠ ⎝1 1⎞ ⎟ 0⎠ ⎛1 ⎜ ⎝0 ⎛1 ⎜ ⎝0 0 ⎞ ⎛ 0 ⎟ ⎜ −1 ⎠ ⎝ −1 0 ⎞ ⎛ 0 ⎟ ⎜ −1 ⎠ ⎝ −1 1⎞ ⎛0 ⎟ ⎜ 0⎠ ⎝1 1⎞ ⎛0 ⎟ ⎜ 0⎠ ⎝1 −1 ⎞ ⎛ 0 ⎟ ⎜ 0 ⎠ ⎝1 −1 ⎞ ⎛ 0 ⎟ ⎜ 0 ⎠ ⎝1 1⎞ ⎟ 0⎠ © Susan Wall, MEI 2006 1⎞ ⎟ 0⎠ 1⎞ ⎟ 0⎠ 1⎞ ⎟ 0⎠ Page 2 of 2 MEI FP1, June 09 ⎛3 0⎞ ⎛0 1⎞ ⎛ 0 −1 ⎞ , N = Q = and ⎟ ⎜1 0⎟ ⎜1 0 ⎟ . ⎝0 2⎠ ⎝ ⎠ ⎝ ⎠ You are given that M = ⎜ 9 (i) The matrix products Q(MN) and (QM)N are identical. What property of matrix multiplication does this illustrate? Find QMN [4] M, N and Q represent the transformations M, N and Q respectively. (ii) [4] Describe the transformations M, N and Q. 6 5 4 3 2 A C 1 -5 -4 -3 -2 -1 O B 1 -1 2 3 4 5 Fig. 9 (iii) The points A, B and C in the triangle in Fig. 9 are mapped to the points A′, B′ and C′ respectively by the composite transformation N followed by M followed by Q. Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of [4] each point clearly. 9(i) 9(ii) 9(iii) Matrix multiplication is associative ⎛ 3 0 ⎞⎛ 0 1 ⎞ MN = ⎜ ⎟⎜ ⎟ ⎝ 0 2 ⎠⎝ 1 0 ⎠ B1 M1 ⎛ 0 3⎞ ⇒ MN = ⎜ ⎟ ⎝ 2 0⎠ ⎛ −2 0 ⎞ QMN = ⎜ ⎟ ⎝ 0 3⎠ M is a stretch, factor 3 in the x direction, factor 2 in the y direction. A1 Attempt to find MN or QM ⎛ 0 −2 ⎞ or QM = ⎜ ⎟ ⎝3 0 ⎠ A1(ft) [4] Stretch factor 3 in the x direction B1 Stretch factor 2 in the y direction B1 N is a reflection in the line y = x. B1 Q is a rotation through 90D about the origin. B1 [4] Applying matrix to points. M1 Minus 1 each error to a minimum of A1 0. ⎛ −2 0 ⎞⎛ 1 1 2 ⎞ ⎛ −2 −2 −4 ⎞ ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ 0 3 ⎠⎝ 2 0 2 ⎠ ⎝ 6 0 6 ⎠ B2 Correct, labelled image points, minus 1 each error to a minimum of 0. Give B4 for correct diagram with no workings. [4] MEI FP1, Jan 06 A transformation T acts on all points in the plane. The image of the general point P is denoted by P′ . 9 Every point P is mapped onto a point P′ on the line y = 2 x . The line joining a point P to its image P′ is parallel to the x-axis. • • (i) Write down the image of the point (10, 50 ) under transformation T. [1] (ii) P has coordinates ( x, y ) . State the coordinates of P′ . [2] (iii) All points on a particular line, l, are mapped onto the point ( 3, 6 ) . Write down the equation of the line l. [1] (vi) In part (iii), the whole of the line l was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines. [1] (vii) For a different set of lines, the transformation T has the same effect as translation parallel to the x-axis. Describe this set of lines. [1] (vi) Find the 2 × 2 matrix which represents the transformation. (vii) Show that this matrix is singular. Interpret this result in terms of the transformation. 9(i) 9(ii) ( 25,50 ) B1 ⎛1 ⎞ ⎜ y, y ⎟ ⎝2 ⎠ B1, B1 [1] [2] 9(iii) y=6 B1 [1] 9(iv) All such lines are parallel to the x-axis. 9(v) All such lines are parallel to y = 2 x . 9(vi) ⎛ ⎜0 ⎜⎜ ⎝0 1⎞ 2⎟ ⎟ 1 ⎟⎠ B1 [1] B1 [1] B3 Minus 1 each error [3] 9(vii) ⎛ 0 det ⎜ ⎜⎜ ⎝0 1⎞ 1 2 ⎟ = 0 ×1 − 0 × = 0 ⎟⎟ 2 1⎠ Transformation many to one. M1 E2 [3] Give 1 mark for partial explanation [3] [3] MEI FP1, Jan 07 ⎛ 3 2⎞ ⎛ 1 −3 ⎞ and N = ⎜ ⎟ ⎟. ⎝0 1⎠ ⎝1 4 ⎠ Matrices M and N are given by M = ⎜ 9 (i) Find M −1 and N −1 . (ii) Find MN and ( MN ) . Verify that ( MN ) = N M . [3] −1 (iii) −1 −1 −1 [6] The result ( PQ ) = Q −1P −1 is true for any two 2 × 2 square, non-singular matrices P and Q. The first two lines of a proof of this general result are given below. Beginning with these two lines, complete the general proof. −1 ( PQ ) −1 ( PQ ) = I −1 ⇒ ( PQ ) PQQ −1 = IQ −1 9(i) 1 ⎛1 M −1 = ⎜ 3⎝0 N −1 = 9(ii) −2 ⎞ 3 ⎟⎠ 1⎛ 4 3⎞ ⎜ 7 ⎝ −1 1 ⎟⎠ ⎛ 3 2 ⎞⎛ 1 −3 ⎞ ⎛ 5 −1⎞ MN = ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ 0 1 ⎠⎝ 1 4 ⎠ ⎝ 1 4 ⎠ ( MN ) −1 1 ⎛ 4 1⎞ 21 ⎜⎝ −1 5 ⎟⎠ = 3 ⎞ 1 ⎛ 1 −2 ⎞ ⎜ ⎟× ⎜ ⎟ 7 ⎝ −1 1 ⎠ 3 ⎝ 0 3 ⎠ 1 ⎛ 4 1⎞ −1 = ⎜ = ( MN ) ⎟ 21 ⎝ −1 5 ⎠ N −1M −1 = 9(iii) 1⎛ 4 [4] M1 A1 A1 [3] Dividing by determinant One for each inverse M1 A1 A1 M1 A1 Multiplication A1 Statement of equivalence to ( MN ) [6] ⇒ ( PQ ) PQQ −1 = IQ −1 −1 ⇒ ( PQ ) PI = Q −1 −1 ⇒ ( PQ ) P = Q −1 −1 ⇒ ( PQ ) PP −1 = Q −1P −1 −1 M1 QQ −1 = I A1 M1 Post-multiply by P −1 ⇒ ( PQ ) I = Q −1P −1 −1 ⇒ ( PQ ) −1 = Q −1P −1 A1 [4] −1 MEI FP1, Jan 09 10 ⎛ 3 4 −1 ⎞ ⎛ 11 −5 −7 ⎞ ⎜ ⎟ ⎜ ⎟ You are given that A = ⎜ 1 −1 k ⎟ and B = ⎜ 1 11 5 + k ⎟ and that AB is of the form ⎜ −2 7 −3 ⎟ ⎜ −5 29 7 ⎟⎠ ⎝ ⎠ ⎝ 4k − 8 ⎞ α ⎛ 42 ⎜ ⎟ AB = ⎜10 − 5k −16 + 29k −12 + 6k ⎟ . ⎜ 0 ⎟ 0 β ⎝ ⎠ (i) Show that α = 0 and β = 28 + 7 k . [3] (ii) Find AB when k = 2 . [2] (iii) For the case when k = 2 write down the matrix A -1 . [3] (iv) Use the result from part (iii) to solve the following simultaneous equations. 3x + 4 y − z = 1 x − y + 2 z = −9 −2 x + 7 y − 3 z = 26 10(i) 10(ii) α = 3 × −5 + 4 × 11 + −1 × 29 = 0 β = −2 × −7 + 7 × ( 5 + k ) + −3 × 7 = 28 + 7 k ⎛ 42 0 0 ⎞ AB = ⎜⎜ 0 42 0 ⎟⎟ ⎜ 0 0 42 ⎟ ⎝ ⎠ 10(iii) A −1 ⎛ 11 1 ⎜ = ⎜1 42 ⎜ ⎝ −5 B1 M1 A1 [3] B2 Minus 1 each error to min of 0 [2] −5 −7 ⎞ 11 7 29 [4] ⎟ ⎟ 7 ⎟⎠ M1 Use of B 1 B1 42 A1 Correct inverse [3] 10(iv) ⎛ 11 1 ⎜ 1 42 ⎜⎜ ⎝ −5 −5 11 29 −7 ⎞ ⎛ 1 ⎞ ⎛ x⎞ ⎜ ⎟ ⎜ ⎟ ⎟ 7 ⎜ −9 ⎟ = ⎜ y ⎟ ⎟ 7 ⎟⎠ ⎜⎝ 26 ⎟⎠ ⎜⎝ z ⎟⎠ ⎛ −126 ⎞ ⎛ −3 ⎞ 1 ⎜ = ⎜ 84 ⎟⎟ = ⎜⎜ 2 ⎟⎟ 42 ⎜ ⎟ ⎜ ⎟ ⎝ −84 ⎠ ⎝ −2 ⎠ x = −3, y = 2, z = −2 M1 Attempt to pre-multiply by A −1 s.c. B1 or B2 (see performance) for Gaussian elimination B3 [4] Minus 1 each error
