## matrix-
A general 3×3 matrix
The first mathematical use of the word "matrix" was around 1850 by James Sylvester who saw a matrix as a way of obtaining determinants, but didn't fully appreciate their potential. Within a year of his first use of the term, he introduced the idea to Arthur Cayley who was the first to publish the inverse of a matrix and to treat matrices as purely abstract mathematical forms. The use of mathematical arrays to solve problems predates the application of the name by about 2000 years. Around 200 BC in the Chinese text*Juizhang Suanshu*(Nine Chapters on the Mathematical Arts) the author solves a system of three equations in three unknowns by placing the coefficients on a counting board and solving by a process that today would be called**Gaussian elimination**. "Matrix" comes from the same Latin root that gives us "mother", and was used to refer to the womb and to pregnant animals. It became generalized to mean any situation or substance that contributes to the origin of something.
**A**is described as an*m*×*n*matrix over*F*, where*F*is a field, without characteristic, to which all of the*mn*elements of**A**belong.**A**is made up of*m*1 ×*n*matrices (row vectors) and*nm*× 1 matrices (column vectors).
The**main diagonal**In the square matrix [*a*_{ij}], the elements*a*_{11},*a*_{22}, ...,*a*_{nn}, running from the top left corner to the bottom right corner.
### Types of matrixA**diagonal matrix**is a matrix that has 0 entries along all nondiagonal entries, i.e., only the main diagonal may have non-zero values.
A**unimodular matrix**is a square matrix whose determinant is 1.
A**Hankel matrix**is one in which all the elements are the same along any diagonal that slopes from northeast to southwest.
A**Jordan matrix**(or Jordan block), is one whose diagonal elements are all equal (and nonzero) and whose elements above the principal diagonal are equal to 1, but all other elements are 0.
A**Toeplitz matrix**is one in which all the elements are the same along any diagonal that slopes from northwest to southeast.
### AdditionIf two matrices**A**and**B**of*F*are both*m*×*n*, the result of their addition is defined to be the*m*×*n*matrix of which a typical element (*a*+_{ij}*b*). Their addition is thus commutative and associative. Moreover, the set of_{ij}*m*×*n*matrices of*F*forms an Abelian group under addition since it has an identity element (see group),**O**, the zero matrix whose elements are all zero, since**A**+**O**=**A**for all**A**, and since every**A**has an inverse = -**A**, whose typical element is (-*a*), such that_{ij}**A**+ (-**A**) =**O**.
### MultiplicationIf*l*is an element of*F*, the product*l***A**is defined as the matrix whose typical element is*la*. Multiplication of one matrix by another is possible only if the first has the same number of columns as the second has rows. If_{ij}**A**is an*m*×*n*and**B**an*n*×*p*matrix then their product**AB**is an*m*×*p*matrix with typical element Notice that, in this case,*a*_{i}_{1}*b*_{1}+_{j}*a*_{i}_{2}*b*_{2}+ ... +_{j}*a*_{in}*b*_{nj}**BA**does not exist unless*p = m*since otherwise**B**does not have the same number of columns as**A**has rows.
### Transposition and symmetryThe transpose of an*m*×*n*matrix**A**is an*n*×*m*matrix**A'**, obtained by setting the row vectors of**A**as the column vectors of**A'**, the column vectors of**A**as the row vectors of**A'**. Some basic results emerge:( where**A'**)**'**=**A**, (**AB**)**'**=**B'A'**, and (*k***A**+*l***B**)**'**=*k***A'**+*l***B'**,*k*,*l*, are elements of*F*. A matrix**A**is termed symmetric when**A**=**A'**and skew-symmetric when**A**= -**A'**: of course, only matrices where*n = m*can be symmetric or skew-symmetric. For symmetry the element*a*in_{ij}**A**must equal*a*for all_{ji}*i*,*j*; for skew-symmetry*a*+_{ij}*a*= 0, for all_{ji}*i*,*j*. See also determinant.
• ALGEBRA - In geology, the matrix is the solid matter in
which a fossil or crystal is embedded. Also, a binding substance (e.g., cement in concrete).
• GEOLOGY AND PLANETARY SCIENCE
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