Why was matrix multiplication defined the way it is?

[quasar987]

What was the (historical) motivation for defining the rules of matrix multiplication the way it is?

 

[masudr]

As far as I know, it comes from performing 2 linear transformations, A and B, one after the other. Then realising that both linear transformations can be written as one matrix, C, and then realising that C can be written as the A and B if we define this new operation called matrix multiplication.

 

[tacman]

Matrix multiplication was defined the way it is in order to accurately represent and manipulate mathematical operations involving matrices. The historical motivation for defining the rules of matrix multiplication can be traced back to the need for efficient computation and representation of linear transformations.

In the late 19th and early 20th century, mathematicians and scientists were facing the challenge of solving systems of linear equations, which arise in various fields such as physics, engineering, and economics. To solve these systems, they needed a way to represent and manipulate large amounts of data in a concise and efficient manner. This led to the development of matrices as a mathematical tool for representing linear transformations.

However, in order to perform operations on matrices, such as addition and multiplication, certain rules needed to be established. For addition, the rules were relatively straightforward - matrices of the same size could be added together by adding corresponding entries. However, when it came to multiplication, it was not as simple.

The historical motivation for defining the rules of matrix multiplication was to accurately represent and compute the composition of linear transformations. This was important in fields such as physics and engineering, where multiple transformations needed to be performed in a specific order to achieve a desired result. The rules of matrix multiplication were carefully chosen to ensure that the result of multiplying two matrices together would accurately represent the composition of linear transformations.

Another motivation for defining matrix multiplication was the need for efficient computation. The rules were designed to minimize the number of operations required to multiply two matrices together, making it a more efficient process. This was especially important in the early days of computing when resources were limited and time-consuming calculations needed to be avoided.

In conclusion, matrix multiplication was defined the way it is in order to accurately represent and manipulate linear transformations, as well as to ensure efficient computation. Its historical motivation can be traced back to the need for solving systems of linear equations and the development of efficient computing methods. The rules of matrix multiplication have stood the test of time and continue to be an essential tool in various fields of mathematics and science.