Matrices

Matrices are a fundamental topic in mathematics, and play a critical role in many fields, including engineering, computer science, economics, and physics.  In the International Baccalaureate (IB) Mathematics curriculum, matrices appear primarily in the Applications & Interpretations Higher Level (AI HL) course.

What are matrices?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.  The size of a matrix is defined by its dimensions, which are expressed as the number of rows and columns.  For example, a matrix with 3 rows and 4 columns is called a 3x4 matrix.  The entries of a matrix can be real numbers, complex numbers, or other mathematical objects, such as polynomials, vectors, or functions.  In IB, however, matrices will usually only contain real number entries.  Matrices are usually denoted by capital letters, such as A, B, C, or X.

Who invented matrices?

We can’t really say that any person in particular ‘invented’ matrices.  Chinese mathematicians used techniques that correspond to modern matrices over 2000 years ago in order to solve equations.  The study of matrices in their modern form is mostly a product of the past 300 years or so, and has included contributions from a large number of preeminent mathematicians.

What does the word ‘matrix’ mean?

‘Matrix’ is the Latin word for ‘womb’, and from that meaning it could also be used to refer to the ‘source’ or ‘origin’ of something.  The word seems first to have been used for mathematical matrices by James Joseph Sylvester in 1850.  Sylvester used the word because he considered a matrix to be the thing that ‘gives birth to’ various quantities called determinants.

Why are matrices important?

Matrices provide a powerful tool for modelling and solving problems that involve multiple variables or equations.  They are especially useful in linear algebra, where they can represent linear transformations, systems of linear equations, and vector spaces.  Matrices also allow us to perform various operations, such as addition, subtraction, multiplication, and inversion, which can simplify complex computations and reveal important properties of the underlying structure.  Furthermore, matrices can be used to represent and manipulate data in many fields, such as image processing, signal analysis, and network theory.

What are matrices used for?

Matrices have numerous applications in mathematics and beyond.  Here are some examples:
In physics, matrices are used to represent rotations, reflections, and other transformations in three-dimensional space.  They are also used to describe the behaviour of quantum systems, such as particles and waves.

In computer graphics, matrices are used to transform and project three-dimensional objects onto a two-dimensional screen.  They are also used to simulate lighting, shading, and other visual effects.
In economics, matrices are used to represent input-output models, where the production and consumption of goods and services are linked through a system of equations.  They are also used to analyse market trends, financial risks, and portfolio optimisation.

In biology, matrices are used to analyse genetic sequences, where the nucleotides are arranged in rows and columns.  They are also used to model ecological systems, such as food webs and population dynamics.

In cryptography, matrices are used to encrypt and decrypt messages, by applying matrix operations to the plaintext and the key.

How do you add or subtract two matrices?

To add or subtract two matrices, we simply add or subtract their corresponding elements.  The matrices must have the same dimensions, otherwise the operation is not defined.

Note that matrix addition and subtraction satisfy the following properties:

• Commutativity: A + B = B + A and A - B ≠ B - A in general.
• Associativity: (A + B) + C = A + (B + C) and (A - B) - C ≠ A - (B - C) in general.
• Identity: There exists a matrix O such that A + O = A and A - O = A for any matrix A.  This matrix is called the zero matrix or null matrix, and its entries are all equal to zero.
• Inverse: For any matrix A, there exists a matrix -A such that A + (-A) = O and (-A) + A = O, where O is the zero matrix.  This matrix is called the negative or opposite of A, and its entries are the negatives of the corresponding entries of A.

How do you multiply two matrices?

To multiply two 2x2 matrices, you can use the following formula:

For matrices of other sizes, see the revision notes on matrix multiplication here at Save My Exams.

Note that matrix multiplication satisfies the following properties:

• Non-commutativity: A x B ≠ B x A in general.
• Associativity: (A x B) x C = A x (B x C) for any matrices A, B

What is the best way to learn how to work with matrices?

The best way to learn matrices is through practice and understanding the underlying concepts.

This includes understanding how to perform operations on matrices, such as addition, subtraction, multiplication, and finding determinants and inverses.  It is important to understand the properties of these operations, such as commutativity, associativity, and distributivity, and how they apply to matrices.

It is also important to become familiar with the terminology associated with matrices, such as rows, columns, dimensions, and identity matrices, and how to interpret them in different contexts.

Another useful approach to learning matrices is to study and solve problems from textbooks, online resources, or past papers, as this will help you develop a deeper understanding of the concepts and how to apply them to various scenarios.  Additionally, working with a teacher, tutor, or study group can be helpful in clarifying any questions or difficulties you may encounter.

Overall, the key to learning matrices is to practise consistently and seek to understand the underlying concepts and properties, as this will not only help you in your IB Mathematics course, but also in other areas of mathematics and beyond.

Where can I start exploring matrices here at Save My Exams?

Here are some good introductory pages on matrix topics:

• Introduction to Matrices
• Operations with Matrices
• Solving Systems of Linear Equations Using Matrices
• Eigenvalues & Eigenvectors