A matrix is a fundamental structure in linear algebra, consisting of a rectangular arrangement of numbers, symbols, or expressions organized in rows and columns. Each position in this arrangement (called an element or entry) is uniquely identified by its row and column location.
For example, a basic 2×3 matrix would be written as:
⎡1 2 3⎤
⎣4 5 6⎦
where the dimensions "2×3" indicate two rows and three columns. This structure forms the cornerstone of linear algebraic operations and transformations.
The historical development of matrices is intertwined with the evolution of linear algebra itself, tracing back to ancient China's work on systems of linear equations. However, the modern concept emerged in the mid-19th century through mathematicians like Arthur Cayley and James Joseph Sylvester, who formalized matrix operations within linear algebra. These operations, including addition, multiplication, and finding determinants, provide the computational framework for solving linear systems and understanding vector spaces – key concepts in linear algebra.
Matrices serve as powerful tools across various scientific and technical fields. In computer graphics, they enable transformations like rotation, scaling, and reflection of objects in space. In quantum mechanics, matrices represent quantum states and operations. Economic models use matrices to analyze complex systems of inputs and outputs, while in machine learning, matrices store neural network weights and enable efficient data transformations. This versatility stems from their ability to compactly represent and manipulate multiple pieces of related information simultaneously.