Matrices, Eigenvalues & Transformations

IB Mathematics · Study Notes

Matrices, Eigenvalues
& Transformations

From basic matrix algebra to diagonalisation, Markov chains, and geometric transformations.

Matrix Algebra and Operations

A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is written as rows × columns.

Basic Operations

Operation	Condition	Notes
Addition / Subtraction	Same order only	Corresponding elements combined.
Scalar Multiplication	Always defined	Every element multiplied by the scalar.
Matrix Multiplication \(AB\)	Cols of \(A\) = Rows of \(B\)	Non-commutative — \(AB \neq BA\) in general.

Determinant & Inverse

Determinant (2×2)

\( \det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad – bc \)

Inverse (2×2)

\( A^{-1} = \dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix} \)

Solving Linear Systems

\( A\mathbf{x} = \mathbf{b} \;\Rightarrow\; \mathbf{x} = A^{-1}\mathbf{b} \)

Singular vs. Non-Singular

Singular\(\det A = 0\) — no inverse exists. Not Invertible
Non-Singular\(\det A \neq 0\) — inverse exists. Invertible

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors reveal the fundamental behaviour of a square matrix — the directions it stretches or flips, and by how much.

Core Definitions

Eigenvalue Equation

\( A\mathbf{x} = \lambda\mathbf{x} \)

Equivalent Form

\( (\lambda I – A)\mathbf{x} = \mathbf{0} \)

Characteristic Equation

\( \det(\lambda I – A) = 0 \)

Finding Eigenvalues & Eigenvectors

Form \(\lambda I – A\) and compute its determinant.
Set \(\det(\lambda I – A) = 0\) — this is the characteristic polynomial.
Solve for \(\lambda\) (the eigenvalues). For a 2×2 matrix, this gives a quadratic.
For each \(\lambda\), substitute back into \((\lambda I – A)\mathbf{x} = \mathbf{0}\) and solve for \(\mathbf{x}\).

Worked Example

Let \( A = \begin{pmatrix} 0 & -3 \\ -4 & -1 \end{pmatrix} \).

Characteristic equation: \[ \det\begin{pmatrix}\lambda & 3 \\ 4 & \lambda+1\end{pmatrix} = \lambda(\lambda+1) – 12 = \lambda^2 + \lambda – 12 = 0 \]

Factoring: \((\lambda – 3)(\lambda + 4) = 0\), so \(\lambda = 3\) or \(\lambda = -4\).

III

Diagonalisation and Powers

A matrix is diagonalisable when it can be decomposed using its eigenvectors into a simpler diagonal form — making repeated operations like \(A^n\) far more efficient.

The Diagonalisation Decomposition

Diagonalised Form

\( D = P^{-1}AP \)

Matrix Power

\( A^n = PD^nP^{-1} \)

How to Construct P and D

Matrix PPlace the eigenvectors of \(A\) as the columns of \(P\).
Matrix DPlace the corresponding eigenvalues on the leading diagonal of \(D\); all other entries are zero.
Why use it?Computing \(D^n\) is trivial — just raise each diagonal entry to the power \(n\). No repeated matrix multiplication required.

Diagonal Power Rule

\( D^n = \begin{pmatrix}\lambda_1^n & 0 \\ 0 & \lambda_2^n\end{pmatrix} \)

Markov Chains and Transformations

Markov Chains

A stochastic model using a transition matrix \(T\) to predict how a system evolves over discrete time steps.

State at Step n

\( \mathbf{s}_n = T^n\,\mathbf{s}_0 \)

Steady State Condition

\( T\mathbf{s} = \mathbf{s} \)

Key Insight

The steady state \(\mathbf{s}\) is an eigenvector of \(T\) with eigenvalue \(\lambda = 1\). The system settles here as \(n \to \infty\) (under suitable conditions).

Geometric Transformations

2×2 matrices can encode geometric transformations of the plane. Common types:

↺

Rotation (anticlockwise)

\( \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix} \)

↔

Reflection in \(y = (\tan\alpha)x\)

\( \begin{pmatrix}\cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha\end{pmatrix} \)

⇄

Composite (A then B)

\( BA \)

Order Matters — Composite Transformations

If transformation \(A\) is applied first and then \(B\), the combined matrix is \(BA\) — note the reverse order. This reflects the non-commutativity of matrix multiplication.

Quick Reference

Transformation	Matrix	Notes
Rotation by \(\theta\)	\(\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}\)	Anticlockwise about origin.
Reflection in \(y=(\tan\alpha)x\)	\(\begin{pmatrix}\cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha\end{pmatrix}\)	e.g. \(y=x\) gives \(\alpha=45°\).
Enlargement by scale \(k\)	\(\begin{pmatrix}k & 0 \\ 0 & k\end{pmatrix}\)	All lengths scaled by \(k\).
Stretch (x-direction) by \(k\)	\(\begin{pmatrix}k & 0 \\ 0 & 1\end{pmatrix}\)	y-coordinates unchanged.

Matrices & Eigenvalues Quiz

IB Mathematics · Practice Quiz

Matrices & Eigenvalues

10 questions · Multiple choice · Instant feedback

Question 1 of 10 0%

Question 1 / 10

For a square matrix \(A\), what is the defined method to find the characteristic polynomial \(p(\lambda)\)?

Question 2 / 10

Given the matrix \(A = \begin{pmatrix}2&5\\1&0\end{pmatrix}\), what are its eigenvalues?

Question 3 / 10

In the context of matrix diagonalisation, if a \(2 \times 2\) matrix \(A\) has distinct eigenvalues \(\lambda_1\) and \(\lambda_2\) with corresponding eigenvectors \(\mathbf{x}_1\) and \(\mathbf{x}_2\), how is the matrix \(P\) constructed such that \(P^{-1}AP = D\)?

Question 4 / 10

Which of the following describes the relationship between the eigenvalues of a matrix \(A\) and the eigenvalues of its power \(A^n\)?

Question 5 / 10

In a Markov chain with transition matrix \(T\) and state matrix \(\mathbf{s}\), what condition must be met for \(\mathbf{s}\) to be considered a ‘steady state’ matrix?

Question 6 / 10

A linear transformation is defined by the matrix \(A = \begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\). What geometric operation does this matrix represent?

Question 7 / 10

If an object with area \(15\) units² undergoes an affine transformation \(\mathbf{x}’ = A\mathbf{x} + \mathbf{b}\) where \(A = \begin{pmatrix}2&1\\-1&3\end{pmatrix}\), what is the area of the image?

Question 8 / 10

What is the primary difference between a linear transformation and an affine transformation?

Question 9 / 10

Which property defines the identity matrix \(I\) in matrix multiplication?

Question 10 / 10

When finding eigenvectors for an eigenvalue \(\lambda\), the system \((\lambda I – A)\mathbf{x} = \mathbf{0}\) always leads to which of the following?

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