The study of matrices is quite old. A 3-by-3 magic square appears in Chinese
literature dating from as early as 650 BC. Matrices have a long history of application in
solving linear equations. After the development of the theory of determinants by Seki
Kowa and Leibniz in the late 17th century, Cramer developed the theory further in the
18th century, presenting Cramer's rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan
developed Gauss-Jordan elimination in the 1800s. Cayley, Hamilton, Grassmann,
Frobenius and von Neumann are among the famous mathematicians who have worked on
matrix theory. Olga Taussky-Todd (1906-1995) used matrix theory to investigate an
aerodynamic phenomenon called fluttering or aeroelasticity during WWII.
In mathematics, a matrix (plural matrices) is a rectangular table of elements (or
entries), which may be numbers or, more generally, any abstract quantities that can be
added and multiplied. Matrices are used to describe linear equations, keep track of the
coefficients of linear transformation and to record data that depend on multiple
parameters. Matrices can be added, multiplied, and decomposed in various ways, making
them a key concept in linear algebra and matrix theory.
1.2.1 Applications of matrices
Matrices can be used to encrypt numerical data. Multiplying the data
matrix with a key matrix does encryption. Simply multiplying the encrypted
matrix with the inverse of the key does decryption.
4×4 transformation matrices are commonly used in computer graphics.
The upper left 3×3 portion of a transformation matrix is composed of the new X,
Y, and Z-axes of the post-transformation coordinate space.
1.2.2 Definition of a Matrix
If m and n are positive integers, then an m ´ n matrix (read “m b y n”) is a
æ a11 a12 a13 . . a1n öü
ç a 21 a 22 a 23 . . a 2 n ÷ï
a32 a33 . . a3n ÷ïï
÷ý m rows
. . . . ÷ï
. . . . ÷÷ï
a n 2 a n 3 . . a nn ÷øïþ
in which each entry, ai,j, of the matrix is a real number. An m ´ n matrix has m rows
(horizontal lines) and n columns (vertical lines).