Further Maths - Solving Systems of Equations Using Matrices
Setting up the matrix equation
What is the crucial idea that allows you to solve systems of equations using matrices?
If
\[A\left( \begin{matrix} x \\\\ y \\ z\end{matrix} \right) = v\]then
\[\left( \begin{matrix} x \\\\ y \\ z\end{matrix} \right) = A^{-1}v\]How could you rewrite
\[2x+3y = 4 \\\\ 4x-y = 7\]
as the product of two matrices?
How could you solve this
\[\left(\begin{matrix}2 & 3 \\\\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}4 \\ 7\end{matrix}\right)\]
?
Multiply both left-hand sides of the equation by the inverse of the matrix.
How could you rewrite
\[2x-6y+4z = 32 \\\\ 3x + 2y -9z = -49 \\ -2x + 4y + z = -3\]
(the gist is fine, not exact)?
Consistency
What does it mean for a system of equations to be consistent?
There is at least one set of values that satisfies all three equations simultaneously.
What does it mean for a system of equations to be inconsistent?
There are no sets of values that satisfy all three equations simultaneously.
Geometric configurations of three planes
What is the geometric visualisation of three planes meeting at a point?
PHOTO
What does it mean in terms of the number of solutions for three planes meeting at a point?
The system of equations is consistent and has only one solution.
If a system of equations is consistent and only has one solution, then the planes form?
A point.
PHOTO
What is the geometric visualisation of three planes forming a sheaf?
PHOTO
What does it mean in terms of the number of solutions and consistency for three planes forming a sheaf?
The system of equations is consistent and has infinitely many solutions along a line.
If a system of equations is consistent and has infinitely many solutions along a line, then the planes form?
A sheaf.
PHOTO
What does it mean in terms of the number of solutions and consistency for three planes being parallel and non-identical?
The system of equations is inconsistent and has no solutions.
What is the geometric visualisation of two or more planes being parallel and non-identical?
PHOTO
If a system of equations is inconsistent and has no solutions, then the planes can form what two scenarios?
- A prism
- Two or more parallel, non-identical planes
What does it mean in terms of the number of solutions and consistency for three planes forming a prism?
The system of equations is inconsistent and has no solutions.
What is the geometric visualisation of two or more planes forming a prism?
PHOTO
What does it mean in terms of the number and solutions and consistency for three identical planes?
The system of equations is consistent and has infinitely many solutions along plane.
What is the geometric visualisation of three planes being the same?
PHOTO
If a system of equations is consistent and has infinitely many solutions along a plane, then the planes form?
A plane. They’re all the same.
PHOTO
Using the determinant
If a system of equations is consistent but has a zero determinant, what are the two possible plane configurations?
- A sheaf
- A plane
Hi! I’m a system of equations. My determinant is zero, and when you try and solve me manually you get GARBAGE. What’s my consistency and plane configurations?
- Inconsistent
- A triangular prism OR Parallel planes
Hi! I’m a system of equations. My determinant is non-zero. What’s my consistency and plane configuration?
- Consistent
- Planes meet at a point
Hi! I’m a system of equations. My determinant is zero and when you try and solve me manually you get $0 = 0$. What’s my consistency and plane configurations?
- Consistent
- A sheaf OR A it’s the same plane