Matrices in games are used extensively. In the context of physics, matrices are used to represent different coordinate spaces. In games, we often combine coordinate spaces; this is done through matrix multiplication. In game physics, it's useful to move one object into the coordinate space of another object; this requires matrices to be inverted. In order to invert a matrix, we have to find its minor, determinant, cofactor, and adjugate. This chapter focuses on what is needed to multiply and invert matrices.

A matrix is a grid of numbers, represented by a bold capital letter. The number of rows in a matrix is represented by i; the number of columns is represented by j.

For example, in a 3 X 2 matrix, i would be 3 and j would be 2. This 3 X 2 matrix looks like this:

Matrices can be of any dimension; in video games, we tend to use 2 X 2, 3 X 3, and 4 X 4 matrices. If a matrix has the same number of rows and columns, it is called a square matrix. In this book, we're going to be working mostly with square matrices.

Individual elements of the matrix are indexed with subscripts. For example, refers to the element in row 1, column 2 of the matrix M.

The transpose of matrix M, written as is a matrix in which every element i, j equals the element j, i of the original matrix. The transpose of a matrix can be acquired by reflecting the matrix over its main diagonal, writing the rows of M as the columns of , or by writing the columns of M as the rows of . We can express the transpose for each component of a matrix with the following equation:

The transpose operation replaces the rows of a matrix with its columns:

Like a vector, there are many ways to multiply a matrix. In this chapter we will cover multiplying matrices by a scalar or by another matrix. Scalar multiplication is component wise. Given a matrix M and a scalar s, scalar multiplication is defined as follows:

We can also multiply a matrix by another matrix. Two matrices, A and B, can be multiplied together only if the number of columns in A matches the number of rows in B. That is, two matrices can only be multiplied together if their inner dimensions match.

When multiplying two matrices together, the dimension of the resulting matrix will match the outer dimensions of the matrices being multiplied. If A is an matrix and B is an matrix, the product of AB will be an matrix. We can find each element of the matrix AB with the following formula:

This operation concatenates the transformations represented by the two matrices into one matrix. Matrix multiplication is not cumulative. . However, matrix multiplication is associative...

Multiplying a scalar number by 1 will result in the original scalar number. There is a matrix analogue to this, the identity matrix. The identity matrix is commonly written as I. If a matrix is multiplied by the identity matrix, the result is the original matrix .

In the identity matrix, all non-diagonal elements are 0, while all diagonal elements are one . The identity matrix looks like this:

Determinants are useful for solving systems of linear equations; however, in the context of a 3D physics engine, we use them almost exclusively to find the inverse of a matrix. The determinant of a matrix M is a scalar value, it's denoted as .The determinant of a matrix is the same as the determinant of its transpose .

We can use a shortcut to find the determinant of a 2 X 2 matrix; subtract the product of the diagonals. This is actually the manually expanded form of Laplace Expansion; we will cover the proper formula in detail later:

One interesting property of determinants is that the determinant of the inverse of a matrix is the same as the inverse determinant of that matrix:

Finding the determinant of a 2 X 2 matrix is fairly straightforward, as we have already expanded the formula. We're just going to implement this in code.