c. 2000–1600 BCE: Babylonian mathematics uses tabular methods to solve systems of linear equations (early precursors to matrix thinking).

c. 300 BCE: Greek mathematics (e.g., Euclid) formalizes geometric methods that later connect to linear relations and transformations.

1693: Gottfried Wilhelm Leibniz explores systematic ways to solve linear systems and introduces ideas related to determinants.

1750: Gabriel Cramer publishes Cramer’s Rule, linking systems of linear equations to determinant-like computations.

1809: Carl Friedrich Gauss popularizes elimination methods (Gaussian elimination) for solving linear systems efficiently.

1812: Augustin-Louis Cauchy advances determinant theory, providing a stronger algebraic foundation for later matrix operations.

1850: James Joseph Sylvester introduces the term “matrix” as an object generating determinants.

1858: Arthur Cayley develops early matrix algebra (addition, multiplication, inverses) and connects matrices to linear transformations.

1870s–1880s: Ferdinand Frobenius deepens the theory of matrices (e.g., matrix rank, canonical ideas), strengthening the abstract framework.

1890s–1910s: Work by Camille Jordan and others clarifies eigenvalues/eigenvectors and Jordan canonical form, shaping how matrices represent linear maps.

1920s–1930s: Linear algebra becomes a standard language in mathematics and physics (notably quantum mechanics), emphasizing matrices as representations of linear operators.

1940s–1950s: Numerical linear algebra grows with early computers; matrices become central to scientific computation.

1960s–1980s: Efficient algorithms (e.g., LU/QR decompositions) and software libraries make matrix methods widely usable in engineering and statistics.

1990s–present: Matrices underpin modern data science and machine learning (e.g., optimization, embeddings, deep learning), as well as graphics, networks, and signal processing.

A matrix is a rectangular array of numbers/symbols arranged in rows and columns, used to represent linear transformations, systems of linear equations, and structured data.