# Who proved Rank Nullity Theorem?

I have been learning about the Rank Nullity theorem and was trying to understand Who came up with the rank nullity theorem? While i did look up on the internet i came up with almost no answers. Some names i came across were Markham and Fiedler , Barrett and Feinsilver. But i dont know the details of it and what exactly ppl did?

• The paper of Barrett and Feinsilver is from 1981 (sciencedirect.com/science/article/pii/0024379581900926) and the paper of Markham and Fiedler is from 1986 (sciencedirect.com/science/article/pii/0024379586901254). Those papers are about structured matrices and are far too recent, sort of like citing a paper from the 1980s when searching for the original work on relativity. The rank-nullity theorem is probably around 100 years old.
– KCd
Feb 12, 2022 at 16:25
• According to the page mathshistory.st-andrews.ac.uk/HistTopics/…, Frobenius defined the rank of a matrix in 1878 and Sylvester defined the nullity of a matrix in 1884. So I would suspect some form of the rank-nullity theorem was formulated by the 1880s, if not earlier in a different nomenclature.
– KCd
Feb 12, 2022 at 16:40
• Wedderburn's Lectures on Matrices (1934) contains a large historical bibliography of linear algebra in the back, the answer could be found by going through those references. But the theory of linear systems of equations is pretty old, tracing the history of any equivalent statement of the rank-nullity theorem might be pretty tough. This would probably make for a good research project. Some secondary sources on the history of linear algebra, looking through the original works (if you can read French and German it will be much easier), looking for any hints, it should be within reach. Feb 12, 2022 at 18:20
• @nwr the primary language of linear algebra back then was determinants, even if people had matrices in mind. It was a transitional period. Read a bit further down on that page to find an excerpt from Bocher's book: "A matrix is said to be of rank $r$ if it contains at least one $r$-rowed determinant which is not zero, while all determinants of order higher than $r$ which the matrix may contain are zero. A matrix is said to be of rank 0 if all its elements are 0. ... For brevity, we shall speak also of the rank of a determinant, meaning thereby the rank of the matrix of the determinant."
– KCd
Feb 12, 2022 at 18:47
• The work of Frobenius in 1878 was titled "On linear substitutions and bilinear forms". It was about the two important roles of matrices: to represent linear maps and bilinear forms. This paper is discussed in Section 7.5 of Hawkins' "The Mathematics of Frobenius in Context". Hawkins describes the goal of the paper as "to fuse the theory of bilinear forms [...] with the symbolic algebra of linear substitutions" and he notes that the term matrix "was not yet a part of Frobenius' mathematical vocabulary". So Frobenius could not write about the "rank of a matrix" even if he used the idea.
– KCd
Feb 12, 2022 at 18:55