Matrix Definitions

Matrices find many applications. Physics makes use of them in various domains, for example in geometrical optics and matrix mechanics. The latter also led to studying in more detail matrices with an infinite number of rows and columns. Matrices encoding distances of knot points in a graph, such as cities connected by roads, are used in graph theory, and computer graphics use matrices to encode projections of three-dimensional space onto a two-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to matrices. The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical mouvements of the 20th century that utilize a square mathematical matrix (music) to determine the pattern of music intervals. - Wikipedia

Unity	Elementary	Symmetric
Diagonal Matrix with main diagonal elements = 1	All elements = 0 except for  a single 1	Square Matrix with aij = aji
Square	Diagonal	Null
Same number of rows and columns	Square Matrix with aij = 0 for all i¹ j	All elements = 0
Singular
Value of determinant = 0

Determinant

Adjoint

Square Matrix where aij is replaced with I, j cofactor multiplied by (-1)i·j. Adjoint A = adj A

Rank

Maximum number of linearly independent columns of the matrix A. Order of the longest nonsingular matrix contained in matrix A. Rank of A = Rank of A'     Rank of A = Rank of A'A   Rank of A = Rank of A A'

Transpose

Matrix where rows and columns are interchanged. Transpose A = A (A')' = A (kA)' = kA' (A+B)' = A'+B' (AB)' = B'A'