4024/22 May/Jun 2022, Question 8: Matrix inverse
Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 9 marks
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The question, in short
Solve matrix equation for A, find p from a determinant then B⁻¹, and describe and draw transformations of shape A on a grid.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Part (a) · 2 marks
Find the 2×2 matrix A satisfying the given matrix equation.
- 13A = matrix with rows (2, 3), (5, 2) − matrix with rows (5, 3), (−4, −1)
- 23A = matrix with rows (−3, 0) and (9, 3)
- 3A = matrix with rows (−1, 0) and (3, 1)
Answermatrix with rows (−1, 0) and (3, 1)
Part (b) · 3 marks
Given det B = 2 with B rows (2, −2) and (4, p), find p and then B⁻¹.
- 1det B = 2p − (−2 × 4) = 2p + 8 = 2
- 2p = −3
- 3B⁻¹ = (1/2) × matrix with rows (−3, 2) and (−4, 2)
Answer(1/2) × matrix with rows (−3, 2) and (−4, 2), or matrix with rows (−1.5, 1) and (−2, 1)
Part (ci) · 2 marks
Describe fully the single transformation mapping shape A onto shape B.
- 1Each vertex moves 4 left and 3 up, e.g. (−1, 2) → (−5, 5)
- 2Translation by column vector (−4, 3)
AnswerTranslation column vector (−4, 3)
Part (cii) · 2 marks
Draw and label shape C, the image of A under the matrix with rows (−2, 0) and (0, −2).
- 1The matrix multiplies each coordinate by −2 (enlargement, scale factor −2, centre O)
- 2(−1, 2) → (2, −4), (2, 2) → (−4, −4)
- 3(2, 1) → (−4, −2), (0, 1) → (0, −2)
- 4Draw and label C with these vertices
AnswerShape C at (2, −4), (−4, −4), (−4, −2), (0, −2)
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