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.
WHAT YOU PROBABLY THINK
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 machinery scales 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 det 5, so M⁻¹ = (1/5) rows (2, −1), (−1, 3).
v = M⁻¹(5, 5): x = (2 × 5 − 1 × 5)/5 = 1, 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 no solutions at all (inconsistent) or infinitely many along a line or plane.
Three planes, read geometrically
Each equation in three unknowns is a plane, and a 3 × 3 system asks where three planes meet. Non-zero determinant: a single point. Zero determinant splits into pictures worth knowing by name. A sheaf: three planes sharing a common line, infinitely many solutions. A prism: planes meeting in pairs along three parallel lines, like a Toblerone, no common point despite no two planes being parallel. That last picture is what defeats the opening claim.
What a matrix cannot move
An invariant point satisfies Mv = v; for most matrices the origin is the only one. An invariant line maps onto itself as a whole: each point may slide along it, but none leaves. Lines through the origin y = mx are tested by feeding (1, m) through the matrix and demanding the image stay on the line, which yields a quadratic in m.
YOUR TURN
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, 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.
THE EXAM BIT
- 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 by 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
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det ≠ 0: solve Mv = b as v = M⁻¹b. det = 0: consistent means sheaf-like, inconsistent means prism or parallel.
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.
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.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.