Matrices Class Log

Grade 12 – Mathematics HL (A&I) - Matrices Class Log

Date: Tuesday 3rd of February 2026

Duration: 45 minutes

Overview

• Focus: Differential equations (coupled systems)

• Homework Correction: Kognity assignment

Differential Equations

Exact Solutions of Coupled Systems of Differential Equations

• Theorem: The solutions of the coupled linear system $\frac{d\mathbf{x}}{dt} = \mathbf{M}\mathbf{x}$ where $\mathbf{x} = C \begin{pmatrix} x \ y \end{pmatrix} D$ can be expressed as: $\mathbf{x} = \mathbf{A} e^{\lambda<em>1 \mathbf{v</em>1} t} + \mathbf{B} e^{\lambda<em>2 \mathbf{v</em>2} t}$

  • Where:

  • $\lambda<em>1$ and $\lambda</em>2$ are the eigenvalues of the matrix $\mathbf{M}$ associated with the eigenvectors $\mathbf{v<em>1}$ and $\mathbf{v</em>2}$

  • Constants $\mathbf{A}, \mathbf{B} \in \mathbb{R}$ depend on initial conditions

• Examination Note: Exact solutions will only be calculated for real distinct eigenvalues.

Phase Portraits

Definition and Description

• Phase Portrait: A representative set of solutions of the coupled system of differential equations
  $\frac{d\mathbf{x}}{dt} = \mathbf{M}\mathbf{x}$

• Parametric Curves:

  • Curves plotted on the Cartesian plane.

  • Illustrate the path of each particular solution over time ($t$).

• Specific Example:

  • Lucia's Mathematical Exploration involved plotting phase trajectories (solution curves) in a phase plane representing the effector cell and tumour cell populations.

• Equilibrium Point: The system stabilizes where both populations remain constant.

  • For equilibrium point $(12, 16)$, it holds that:

  • $\frac{dE}{dt} = 0$

  • $\frac{dT}{dt} = 0$

• Components of Phase Portraits:

  • Consists of equilibrium points and a selection of phase trajectories, demonstrating how variable values change over time.

Properties of Eigenvalues and Eigenvectors

Behaviour of the Coupled System

• Distinct Eigenvalues: When the matrix representing the coupled system has distinct eigenvalues, several observations can be made:

  • Lines through the origin along the direction of eigenvectors indicate gradients for long-term behaviour as $t \rightarrow +\infty$ and $t \rightarrow -\infty$.

  • Positive and Negative Eigenvalues:

  • Negative Eigenvalue ($\lambda$):

    • Solutions are parallel to the eigenvector $\mathbf{v}$ and are directed towards the origin as $t$ increases.

  • Positive Eigenvalue ($\lambda$):

    • Solutions are parallel to $\mathbf{v}$ and move away from the origin as $t$ increases.

  • Mixed Signs:

    • With one positive and one negative eigenvalue, the solution demonstrates a combined behaviour:

    • For large negative $t$, the curve aligns with the negative eigenvalue’s solution, moving towards the origin.

    • For large positive $t$, it aligns with the positive eigenvalue’s solution, moving away from the origin.

Example of a Coupled System of Differential Equations

• System:
  $\frac{dx}{dt} = x \qquad \frac{dy}{dt} = -y$
  This can be expressed as:
  $\begin{pmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{pmatrix} = \begin{pmatrix} 1 &amp; 0 \ 0 &amp; -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$

• Eigenvalues:

  • $\lambda<em>1 = 1$ and $\lambda</em>2 = -1$

• Eigenvectors:

  • $\begin{pmatrix} 1 \ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \ 1 \end{pmatrix}$

• General Solution:
  $\begin{pmatrix} x \ y \end{pmatrix} = \mathbf{A} e^{t} \begin{pmatrix} 1 \ 0 \end{pmatrix} + \mathbf{B} e^{-t} \begin{pmatrix} 0 \ 1 \end{pmatrix}$, where $t \in \mathbb{R}$.

• Long-Term Behaviour:

  • For large negative $t$, solutions approach the origin along the $y$-axis.

  • For large positive $t$, solutions diverge along the $x$-axis.

• Equilibrium Point:

  • A point where both derivatives equal zero:

  • $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} = 0$

• Stability Conditions:

  • Stable Equilibrium Point:

  • All nearby points converge to this point (conditions met when both eigenvalues are negative).

  • Unstable Equilibrium Point:

  • All nearby points diverge from this point (conditions met when both eigenvalues are positive).

  • Saddle Point:

  • One positive and one negative eigenvalue indicates that trajectories approach the origin and subsequently turn away, leading to instability.

Further Example of a Coupled System of Differential Equations

• Coupled System:
  $\begin{aligned} \frac{dx}{dt} &amp;= -x + 9y \ \frac{dy}{dt} &amp;= 6x - 4y \end{aligned}$

• Matrix Representation:
  $\begin{pmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{pmatrix} = \begin{pmatrix} -1 &amp; 9 \ 6 &amp; -4 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$

• Eigenvalues:

  • $\lambda<em>1 = 1$ and $\lambda</em>2 = -2$

• Eigenvectors:

  • $\begin{pmatrix} 3 \ 2 \end{pmatrix}$ and $\begin{pmatrix} 1 \ -1 \end{pmatrix}$

• General Solution:
  $\begin{pmatrix} x \ y \end{pmatrix} = \mathbf{A} e^{t} \begin{pmatrix} 3 \ 2 \end{pmatrix} + \mathbf{B} e^{-2t} \begin{pmatrix} 1 \ -1 \end{pmatrix}, \; t \in \mathbb{R}$

• Behaviour as $t$ Approaches Infinity:

  • For large negative $t$, curves follow $\begin{pmatrix} 1 \ -1 \end{pmatrix}$ (towards origin).

  • For large positive $t$, curves follow $\begin{pmatrix} 3 \ 2 \end{pmatrix}$ (away from origin).

• Equilibrium Point Analysis:

  • The origin $(0, 0)$ is classified as a saddle point.

Additional Exercise and Resources

• Exercise 20.4.1:

  • Complete questions (a) to (d) on pages 881-882.

• Applet for Phase Portrait Verification:

  • Utilize the applet available at:
    GeoGebra Applet to verify the phase portrait.

• Slope Field Diagram:

  • A visual representation of the coupled system of differential equations defined as:
    $\begin{aligned} \frac{dx}{dt} &amp;= -y + x \times (x^2 + y^2) \times \sin\left(\frac{\pi}{x^2 + y^2}\right) \ \frac{dy}{dt} &amp;= x + y \times (x^2 + y^2) \times \sin\left(\frac{\pi}{x^2 + y^2}\right) \end{aligned}$

  • Reference for visual representation: