Maths › Matrices › Systems of equations and invariance
Systems of equations and invariance
Simultaneous equations solved by one inverse, three planes read like a geometry problem, and the lines a transformation cannot move.
Builds on Determinants and inverses.
IN THIS TOPIC
- Write simultaneous equations as a matrix equation and solve with an inverse.
- Interpret consistent and inconsistent three-plane systems geometrically.
- Find invariant points and invariant lines of a matrix transformation.
COMMON MISCONCEPTION
If three planes have no common point, at least two of them must be parallel.
Solving by inverse
A pair of simultaneous equations is really one matrix equation, Mv = b, and if det M ≠ 0 the solution is a single multiplication, v = M⁻¹b. The same method extends to three equations in three unknowns unchanged.
WORKED EXAMPLE
Two equations, one multiplication
Solve 3x + y = 5 and x + 2y = 5 using an inverse matrix.
M has rows (3, 1), (1, 2) and determinant 5, so M⁻¹ = (1/5) times the matrix with rows (2, −1), (−1, 3).
v = M⁻¹(5, 5), giving x = (2 × 5 − 1 × 5)/5 = 1 and y = (−5 + 15)/5 = 2.
Substituting back: 3 + 2 = 5 and 1 + 4 = 5. Both hold.
A zero determinant means no unique solution, and which way it goes depends on the right-hand side. Either there are no solutions at all, or infinitely many along a line or plane.
Three planes, read geometrically
Each equation in three unknowns is a plane, so a 3 × 3 system asks where three planes meet. Non-zero determinant gives a single point. Zero determinant splits into pictures worth knowing by name. A sheaf has three planes sharing a common line, so there are infinitely many solutions. A prism has the planes meeting in pairs along three parallel lines, like a Toblerone, with no common point despite no two planes being parallel.
What a matrix cannot move
An invariant point satisfies Mv = v, and for most matrices the origin is the only one. An invariant line maps onto itself as a whole, so each point may slide along it but none leaves. To test a line y = mx through the origin, feed (1, m) through the matrix and demand that the image still satisfies y = mx, which yields a quadratic in m.
GUIDED PRACTICE
Two invariant lines
For the matrix with rows (2, 1), (1, 2), show that y = x and y = −x are invariant lines, and identify which is a line of fixed points.
Show the working
Take (1, 1) on y = x. The image is (3, 3), still on y = x, three times as far out. The line is invariant and stretched by 3.
Take (1, −1) on y = −x. The image is (1, −1), unmoved. Every point of y = −x is fixed, so it is a line of invariant points, the stronger property.
A line of fixed points is always an invariant line. The converse fails, as y = x shows.
ASSESSMENT FOCUS
- Set up Mv = b explicitly before inverting. The layout carries a mark of its own.
- With det = 0, test consistency before describing the geometry. Sheaf and prism differ only in the right-hand sides.
- Distinguish 'invariant line' from 'line of invariant points', and say which you have shown.
- For lines y = mx + c with c ≠ 0, substitute a general point and match both coordinates.
CHECK YOURSELF
Solve the system 2x + y = 4, x + 3y = 7 by inverse matrix, and verify the answer.
Show a hint
The determinant is 5.
Show the answer
M has rows (2, 1), (1, 3) and determinant 5, so the inverse is (1/5) times rows (3, −1), (−1, 2). Then x = (3 × 4 − 7)/5 = 1 and y = (−4 + 14)/5 = 2. Check: 2 + 2 = 4 and 1 + 6 = 7.
det ≠ 0 solves Mv = b as v = M⁻¹b; det = 0 means the planes miss or share a line.
Invariant lines survive the matrix as sets; lines of fixed points survive point by point.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the systems of equations and invariance questions page.
CHECK YOUR PROGRESS
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- Write simultaneous equations as a matrix equation and solve with an inverse.
- Interpret consistent and inconsistent three-plane systems geometrically.
- Find invariant points and invariant lines of a matrix transformation.
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