History
When looking back at history, the Babylonians were the first to study simultaneous linear equations, but the Chinese between 200 BC and 100 BC came much closer to our idea of a matrix (O'Connor & Robertson 1996). The first use of matrices can be found in the book Nine Chapters on the Mathematical Art. It describes a problem that involves bundles of corn and uses a matrix with coefficients from several linear equations. The author then uses what is now referred to as Gaussian elimination to solve the problem (O'Connor & Robertson 1996). Gaussian elimination is a way to convert a matrix such that the main diagonal numbers are 1's and below the diagonal are 0's. Here is an example:
Also, the first example of a determinant was shown in 1683 by a man named Seki from Japan in the book Method of solving the dissimulated problems (O'Connor & Robertson 1996). In the book, Seki computes the determinants of 2X2, 3X3, 4X4, and 5X5 matrices. This was all before determinants would appear in Europe, which was in 1693 when a man named Leibniz used it in a letter describing a solution for a system of linear equations (O'Connor & Robertson 1996).
Applications
Matrices have a variety of applications. The first application we will look at is how matrices are used in video games. Video games use matrices to animate figures that we can control in the game (Yip 2001). They do this by manipulating points with matrix properties like translations, rotations, and scaling. For example, when looking at this image from Yip,
we can see this is a picture of a plane in a video game. For this plane to move across the sky in the game, it will use translation. In this case, it's important to understand the properties of matrices because we don't want to accidentally distort the shape of the plane. To visualize this further, lets look at another example from the website Matrix Playground:
In this image, the original plane is the black F, while the transformed plane is the pink F. To achieve this transformed plane, the matrix
was applied to the black F's plane. This is another example of translation. If we wanted to rotate the plane like this,
We would need to have a matrix with this general form
| cosθ |
-sinθ |
0 |
| sinθ |
cosθ |
0 |
| 0 |
0 |
1 |
In this same way, video games use matrices to move computer generated planes that players can control in a virtual world.
Another way matrices are used is in the field of cryptography. This means matrices can be used to make secret codes. One method of doing this is called the Hill Algorithm (Sekhon & Bloom 2022). Sekhon and Bloom describe the process like this: first convert a secret message into a string of numbers by randomly assigning a value to each letter. Then convert that string of numbers into a new set of numbers by multiplying the string by a square matrix that has an inverse. This is our new coded message. To decode the message, we multiply the string of numbers by the inverse of our matrix and bring back our letters. However, it should be noted this method is no longer viable do to our current capabilities with computers. The algorithm was created in 1929 afterall (Sekhon & Bloom 2022). Despite this, matrices are still used in current cryptography.
If you want to try the Hill Algorithm, try to decipher this message: "Hoang ecw hvz qeetbzf ig eilduncB." If you are having trouble, you can use the website provided in this link or make your own cipher.
