Regarding the importance of the "rank" of a matrix, and help to understand what this is, intuitively, you'll find some wonderful answers here, to a previous post at math.se:
A wonderful website for origins of mathematical terms is found here:
Click on "[R]"...scroll down to:
"Rank (of a matrix or determinant)":
RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word Rang in his paper "Uber homogene totale Differentialgleichungen," J. reine angew. Math. Vol. 86 (1879) p.1. This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933). Frobenius was defining the rank of a determinant but the term travelled.
In English, rank (of a matrix) is found in the monograph "Quadratic forms and their classification by means of invariant factors", by T. J. Bromwich, Cambridge UP, 1906. This citation was provided by Rod Gow, who writes that it is possible that an earlier book c. 1900 by G. B. Mathews, a revision of R. F. Scott's 1880 book on determinants, contains the word.
Rank is also found in 1907 in Introduction to Higher Algebra by Maxime Bôcher: where it is defined in the same way as in Frobenius:
Definition 3. 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.