## Matrices

### 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 

for_MATRICES-AND-DETERMINANTS