Matrices

Matrices and determinants are algebraic tools used in many areas such as social science, economic and biological weapons, with which it facilitates the management, data manipulation and characterization.




One way to represent matrices by square brackets is as follows:

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Types of matrices

• Symmetric matrix: square matrix where the values of ai,j=aj,i


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• Square Matrix: this matrix where the number of rows is equal to the number of columns.


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• Transpose of a matrix:

The transpose of a m by n matrix is defined to be a n by m matrix that results from interchanging the rows and columns of the matrix. The transpose of a matrix is designated by the superscript T or " ' ". The matrix A and the transpose of a matrix A are as follows.

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• Triangular matrix:

There are two types of triangular matrices: upper triangular matrix and lower triangular matrix. Triangular matrices have the same number of rows as they have columns; that is, they have n rows and n columns. A matrix U is an upper triangular matrix if its nonzero elements are found only in the upper triangle of the matrix, including the main diagonal; that is:
uij = 0 if i > j
A matrix L is an lower triangular matrix if its nonzero elements are found only in the lower triangle of the matrix, including the main diagonal; that is:
lij = 0 if i <>


from:http://publib.boulder.ibm.com/infocenter/clresctr/vxrx/index.jsp?topic=/com.ibm.cluster.essl43.guideref.doc/am501_trimat.html

• Augmented matrix:
This matrix is obtained by combining two matrices, as follows:


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• Band matrix:
Is a matrix with non-zero values are confined in an environment of the main diagonal, forming a band of non-zero values which complement the main diagonal of the matrix and more diagonal in each of its sides.



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• Matrix multiplication:
In matrix multiplication should be noted that the number of columns of a matrix is equal to the number of rows from the other parent to multiply, as:

Am*n * Bn*d

is denoted

AB = Cm*d

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Determinant

The determinant is an algebraic operation that transforms a square matrix M into a scalar. This operation has many useful and important properties. For example, the determinant is zero if and only the matrix M is singular (no inverse exists).

Let M be an n n matrix with entries Mij that are elements of a given field. The determinant of M , or detM for short, is the scalar quantity.



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Bibliografy:
• http://personal.redestb.es/ztt/tem/t6_matrices.htm
• http://portales.educared.net/wikiEducared/index.php?title=M%C3%A9todo_de_reducci%C3%B3n_de_Gauss
• http://planetmath.org/encyclopedia/Determinant2.html