## Introduction

The usefulness of a matrix in computer graphics is its ability to convert geometric data into different coordinate systems. A matrix is composed of elements arranged in rows and columns. In simple terms, the elements of a matrix are coefficients that represents the scale or rotation a vector will undergo during a transformation.

## Matrix

A matrix is an entity composed of components arranged in rows and columns. Mathematically, a matrix is represented as:

The rows and columns of a matrix determines the dimension of a matrix. A matrix containing 2 rows and 3 columns is of dimension 2x3. Here is an example of matrices with different dimensions:

Dimensions in matrix arithmetic is very important, since some operations are not possible unless matrices have identical dimensions. Matrices can be added and subtracted. They can be multiplied by a scalar and multiplied among themselves. Matrices can not be divided, instead a matrix called the **Inverse** is calculated which serves as the reciprocal of the matrix. Matrices have an arrangement property. They can be arranged in **row-major** format or **column-major** format. This is very important to keep in mind since multiplying vectors or matrices of wrong format will result in wrong calculations. OpenGL requires all matrices to be in column-major format

## Addition/Subtraction

Matrix addition/subtraction is allowed between matrices that have the same dimension. To add matrices simply add the corresponding component to each other.

For example:

## Scalar Multiplication

Scalar multiplication is performed by multiplying a scalar with each corresponding component in the matrix.

For example:

## Multiplication

Unlike matrix addition, matrix multiplication does not require matrices to be of the same dimensions. However, the number of rows in matrix **M** must be equal to the number of colums in matrix **N**. For example:

For example:

Note, an important propertry of matrices:

Thus, be careful when multiplying matrices. The order of multiplication matters.

## Identity Matrix

The *Identity Matrix* is a special kind of matrix which is similar to the concept of “1” in real numbers. Just like a real number multiplied by “1” results in the real number itself, any matrix multiply by the *identity matrix* results in the matrix itself.
The *Identity matrix* is defined as:

Multiplying any matrix by the *identity matrix* results in the same matrix.

## Inverse

In arithmetic, dividing a number by “4” is the same as multiplying a number by the inverse of “4”, i.e., “1/4”. Division operation does not exist in matrix arithmetic. However, it is possible to multiply a matrix by an *inverse*. The inverse of a matrix is denoted as:

When a matrix is multiplied by its *inverse*, it produces an *identity matrix*. Similar to when the real number “4” is multiplied by its inverse “1/4” produces “1”.

Keep in mind that not all matrices have an inverse.

## Transpose

A *Transpose* operation takes each row of a matrix and converts it to a corresponding column. Mathematically, a transpose is represented as:

For example, the transpose of matrix M is done as follows:

Why would you need to *transpose* a matrix? A matrix can be represented as either a *row-major* or *column-major* matrix. During **transformation** operations, a vector and a matrix must be in either the same *row-major* or *colum-major* format. If they are not, the matrix must be *transpose*, so the results of the transformation are correct.