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Systems of equations and invariance questions
Simultaneous equations solved by one inverse, three planes read like a geometry problem, and the lines a transformation cannot move.
7 original questions · 31 marks · the systems of equations and invariance notes · Matrices
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Write the simultaneous equations 2x + y = 7 and x + 3y = 11 as a single matrix equation Mv = b, identifying M, v and b.
Worked answer
M has rows (2, 1) and (1, 3), v is the column (x, y) and b the column (7, 11). B1 B1 B1 for M, v and b. One matrix equation carries both scalar equations at once.Using an inverse matrix, solve 2x + y = 7 and x + 3y = 11.
Worked answer
det M = 6 − 1 = 5, so M⁻¹ = (1/5) × rows (3, −1), (−1, 2). Then v = M⁻¹b = (1/5)(21 − 11, −7 + 22) = (2, 3). Check: 2 × 2 + 3 = 7 and 2 + 9 = 11. M1 for det M = 5, A1 for the inverse, M1 for v = M⁻¹b, A1 for (2, 3).Three planes have no common point, yet no two of them are parallel. Describe the two geometric configurations consistent with this, and state which one admits pairwise intersection lines that are parallel to each other.
Worked answer
Either a prism, where the three pairwise intersection lines are parallel and the planes fence off a triangular tube, or two planes meeting in a line that the third misses everywhere by being parallel to it. The prism is the configuration with three parallel intersection lines; both give an inconsistent system with a singular coefficient matrix. B1 B1 for the two configurations, B1 for naming the prism, B1 for the singular coefficient matrix.Show that the line y = x is invariant under the transformation with matrix rows (4, 1) and (2, 3), and find the factor by which points on it are stretched.
Worked answer
Take (1, 1) on the line. It maps to (5, 5), still on y = x, and any multiple (t, t) maps to (5t, 5t) by linearity. The line maps into itself with stretch factor 5. M1 for mapping a general point of the line, A1 for the image (5t, 5t), A1 for the conclusion, B1 for the stretch factor 5. This is an invariant line rather than a line of fixed points. Every point on it moves, but none of them leaves the line.Find the second invariant line through the origin of the transformation with matrix rows (4, 1) and (2, 3).
Worked answer
Try y = mx. The image of (1, m) is (4 + m, 2 + 3m), which lies on y = mx when 2 + 3m = m(4 + m), so m² + m − 2 = 0, giving m = 1 or m = −2. The new line is y = −2x, stretched by factor 2, since (1, −2) maps to (2, −4). M1 for trying y = mx, A1 for m² + m − 2 = 0, A1 for y = −2x, B1 for the stretch factor 2.Explain the difference between an invariant line and a line of invariant points, and give a transformation with an example of each.
Worked answer
On an invariant line each point maps to a point of the same line, possibly elsewhere; a line of invariant points maps every point to itself. For a reflection, the mirror line is a line of invariant points, while any line perpendicular to the mirror is invariant without its points being fixed. B1 B1 for the two definitions, B1 B1 for the mirror line and a perpendicular line as the two examples.The system x + y + z = 1, x + 2y + 3z = 4 and 2x + 3y + kz = 5 has a coefficient matrix M. Find the value of k for which M is singular. For that value of k, determine whether the system is consistent, and describe the geometry of the three planes.
Worked answer
Expanding along the top row, det M = (2k − 9) − (k − 6) + (3 − 4) = k − 4, so M is singular when k = 4. For any other k the inverse exists and there is a single point of intersection. With k = 4 the third equation reads 2x + 3y + 4z = 5, which is exactly the sum of the first two, so it carries no new information and the system is consistent. Solving the first two, subtracting gives y + 2z = 3, so y = 3 − 2z and then x = 1 − y − z = −2 + z. Writing z = λ, the solutions form the line r = (−2, 3, 0) + λ(1, −2, 1). Geometrically the three planes form a sheaf, meeting in that common line. M1 for expanding the determinant, A1 for det M = k − 4, A1 for k = 4, M1 for testing consistency, A1 for the third equation being the sum of the first two, M1 for solving the first two, A1 for the line of solutions, B1 for the sheaf. A singular matrix rules out a unique solution but says nothing about consistency; the sheaf here and an inconsistent prism both give det M = 0, so the equations still have to be tested.
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