Matrices and their many Applications - James Tanton
How do we find large powers of matrices? Where does the idea behind the solution come from?
On this page you can find a full course of videos, created by Dr James Tanton on the below:
The Story of Matrix Diagonalization: Matrices and eigenvalues and vectors
What does it mean to diagonalize a matrix and why would one ever want to? What motivated this task and how does it bring insight and understanding to general matrix theory? We will explore and answer these questions, and see a natural motivation for wanting to find eigenvalues and eigenvectors.
Eigenvalues, Eigenvectors, and Markov Processes.
What intuition and insights do the eigenvalues and eigenvectors of a matrix provide? We will probe deeper into this topic and explore—and fully explain—the behaviour of two-dimensional Markov processes as a lovely application.
Prerequisite Knowledge: The videos do presume familiarity with the basic arithmetic of matrices and the use of 2x2 matrices to represent certain geometric transformations of the plane (included below).
Matrix Theory for Analysing Coupled Linear Differential Equations
The Mathematical Set-Up and Story: Diagonalization, Eigenvalues and vectors, complex numbers and one-dimensional differential equations
A brief review of diagonalization, eigenvalues and eigenvectors, complex numbers, and one-dimensional differential equations towards a means for fully solving systems of differential equations. 
Matrix Theory: Analysing Coupled Linear Differential Equations
Analysing Coupled Linear Differential Equations
Analysing solutions to systems of equations, understanding phase diagrams, and fully explaining long-term behaviour.
Matrix Theory: Analysing Coupled Linear Differential Equations
Crash course: arithmetic of matrices and geometric transformations of the plane
Click on the this link icon to watch a series of in-depth videos covering the key "pre-requisite knowledge" required: what matrices are, why they are useful and how we calculate with them. Calculating a matrix's inverse and solving simultaneous equations using matrices.
How/why, conceptually, do matrices define transformations. An in-depth look at the geometrical meaning of a matrix's determinant and an introduction to Adjacency matrices follow at the end. Matrix Arithmetic, Inverse & Sim Equations, Transformations etc
Matrices, eigenvalues & powers1: The Problem of Large Powers
A Fuller Understanding (not specifically in the syllabus) - other problems that use a function and its inverse to find a solution
Click no the "eye" icon 
below to view this video.
Matrices, eigenvalues & powers2: Diagonalization
Mathematicians like to work very very hard to work out ways to avoid hard work.
Is there a way to convert any matrix into into a diagonal matrix? If so this would make finding powers of matrices much easier!
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A Fuller Understanding: The Geometric Meaning of Eigenvectors
(Optional) Two rotations about the origin - in 2D and 3D!
Practice questions
Eigenvalues and vectors, Diagonalization and Large Powers
SOLUTIONS to Eigenvalues and vectors, Diagonalization and Large Powers Practice Questions
Optional Extra: Defining a plane - matrices and their eigenvalues & vectors
The general theory of linear algebra gives the following fact: If a two-by-two matrix has two distinct eigenvalues, then its corresponding eigenvectors form a “basis” for the plane. This means that any point in the plane can be represented as a (unique) combination of the two eigenvectors.
Student notes sheet (subscription required)
Student's Notes sheet, to accompany the above Matrices, eigenvalues & powers2: Diagonalization video, with space for student's working out.
Exam questions (subscription required for access)
Eigenvalues and vectors, Diagonalization and Large Powers : 50 marks worth of questions
SOLUTIONS to Eigenvalues and vectors, Diagonalization and Large Powers Practice Questions