Matrix and Determinant

You can free download topic Matrix and Determinant   in pdf format. DETERMINANT                                                           ...

You can free download topic Matrix and Determinant  in pdf format.

DETERMINANT                                                                                                                                                                                                                                                                      In linear algebra , the Determinant is a scalar value that can be computed from the element of a square Matrix and encodes certain properties of the linear transformation describes by the Matrix .The determinant of the MatrixA  IS detenoted by det(A), det A,or |A|.Geometrically,it can be viewed as the volume scaling factor of the linear transformation desbcribed by the matrix.This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matricx.the determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.

In the case of a 2 by 2 matrix the determinant may be defined as

                 |A| = 丨a  b 丨                             

|                                ।   c    d  ।  = ad -bc

Each determinant of a 2 x 2 matrix in this mathematics .For example ,a matrix is often used to represent the Coefficients in a system of linear equations and the determinant can be used to solve those  equations although other method of solution are much more computationally efficient .In linear algebra ,a matrix (with entries in a field ) is singular (not invertible ) if and only if its determinant is zero.this leads to the use of determinant in defining the characteristics polynomial of a matrix ,whose roots are the eigenvalues.

In analytic geometry , determinant express the signed n-dimensional volumes of n-dimensional parallelepiped.This leads to the use of determinants in Calculus .The JACOBIAN Determinant in the charge of the variables rule for the integrals of functions of the several variables.Determinants appear frequently in algebraic identities such as the Vandermonde  Identity.

Determinant possess many algebraic properties .One of them is mutiplicativity ,namely that the determinant of a product of matrices is equal to the product of determinant .special types of matrices have special determinant ;for example ,the determinant of an orthogonal matrix

is always plus or minus one,and the determinant of a complex Hermitian matrix is always real .