Matrices

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Important Formulas :

1.A set of numbers (real or complex) arranged in the form of a rectangular array having rows and columns is called a matrix of order . An matrix is usually written as


2.A matrix is called a rectangular matrix if .


3.A matrix is said to be zero matrix or null matrix, if every element of is equal to zero (i.e.,) An zero matrix is denoted by


4.A square matrix is called a digonal matrix if .


5.A square matrix is called a scalar matrix if


6.A square matrix is called an identify matrix or unit matrix if . We denoted the unit matrix of order by or .


7.A square matrix is called an upper triangular matrix if


8.A square matrix is called a lower triangular matrix if


9.The elements of a square matrix for which ,(i.e.,) the elements are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal of the matrix.


10.A diagonal matrix of order having as diagonal elements is denoted by diag


11.If is a square matrix then the sum of elements in the principal diagonal of is called trace of and is denoted by tr i.e., tr


12.If and are two matrices of the same order then their sum is denoted by and is defined by .Also addition is not defined for matrices of different orders.


13.Matrix addition is commutative (i.e.,) if and are matrices of the same order, then .


14.Matrix addition is  associative (i.e.,)   and are matrices of the same order, then .


15.The null matrix is the identity element for matrix addition (i.e.,) .


16.Cancellation laws hold in case of addition of matrices (i.e.,) if  . are matrices of the same order, then ((left cancellation law))( (right cancellation law))


17.Let    be a matrix and the letter be any number called a scalar. Then the matrix obtained by multiplying every element of by is called the scalar multiple of by and is denoted by thus


18.If  is an matrix, then the matrix of the same order such that is called the additive inverse of A. It denoted by . (i.e.,)


19.For two matrices and of the same order, we define .


20.If    are scalars, and are matrices of the same order, then:


21.If    are scalars, and are matrices of the same order, then:


22.If    are scalars, and are matrices of the same order, then:


23.If    are scalars, and are matrices of the same order, then:


24.If    are scalars, and are matrices of the same order, then:


25.Two matrices , and   are conformable for the product ,    if the number of columns in ,   is same as the number of rows in ,   and . Thus, if   and are two matrices; then their product is defined to be the matrix where .


26.If  is matrix and  is  matrix, then  is matrix.


27.Matrix multiplication is associative (i.e.,) whenever both sides are defined.


28.Matrix multiplication is distributive over matrix addition. (i.e.,) and, whenever both sides of equality are defined.


29.If  is , then .


30.If   and are two square matrices of order then:


31.If  and are two square matrices of order then:


32.If  and are two square matrices of order then:


33.If  and are two square matrices of order then:


34.If  and are two square matrices of order then:


35.If  and are two square matrices of order then:


36.The matrix obtained from any given matrix , by interchanging its and columns is called the transpose of the matrix . we denote it by or . i.e.,


37.If  and are two matrices of same type, then .


38.If is any matrix and is any scalar, then .


39.If  and are two matrices which are conformable for multiplication, then .


40.If are matrices  which are conformable for multiplication, then


41.A square matrix is called symmetric matrix if


42.A square matrix is called skew- symmetric matrix if . is a skew- symmetric matrix


43.If is a square matrix of order , then the number is called the determinant of .det


44.If are square matrices of the same order then .


45.A square matrix is said to be singular matrix if det and a non-singular matrix if det


46.For given square matrix , if there exists a square matrix such that ,then is called the multiplicative inverse of


47.If are square matrices of the same order satisfying , and then


48.If a matrix has inverse, it is unique, we denote the inverse of by and


49.If exists for square matrix , then we say that is invertible.


50.If exists for a square matrix , then exists and


51.If and are two invertible matrices of same order, then is also invertible and .


52.If are invertible matrices of same order, then is also invertible and


53.If is an invertible matrix then is also invertible and


54.If is a non - singular matrix, then


55.If is a non - singular matrix, then


56.If is a non - singular matrix, then


57.If is a non - singular matrix, then


58.If is a non - singular matrix, then


59.If is a non - singular matrix, then


60.If is a non - singular matrix, then


61.If is a non - singular matrix, then for any scalar


62.If is a non - singular matrix, then


63.If is a non - singular matrix, then


64.If is a non - singular matrix, then , where


65.If is a non - singular matrix, then where


66.If is a non - singular matrix, then where


67.If and are two non singular matrices of the same order, then


68.If and are two non singular matrices of the same order, then


69.If and are two non singular matrices of the same order, then

 

 


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