Matrices

Definition and formulas

Introduction:- A matrix is an ordered rectangular array of numbers or functions and that numbers and functions are called the elements or the entries of the matrix.

Notion:- Generally matrix is denoted by capital letters and elements are denoted by small letters. The following example showed some matrices:-

               and so on.

⇒ Horizontal lines of elements are called rows of the matrix.

⇒ Vertical lines of elements are called columns of the matrix.

Order of Matrices:- A matrix having m rows and n columns then the order of the matrix is mxn.For Ex:-

             

In both examples first, count the no. of rows that are m and then count the no. of columns that is the value of n.

Then, Order of A is 4x3 and it is also denoted by A= [aij]4x3.

(i.e. i and j denote respectively row and column number ex:- a13=3)

Order of B is 3x3 and it is also denoted by B= [bij]3x3.

Equality of matrices.:- Two matrices A=[aij] and B=[bij] are said to be equal if

(i) they are of the same order.

(ii) each element of A is equal to the corresponding element of B, that is aij= bij for all i and j.For Ex:-

  are equal matrices.

are not equal matrices because corresponding elements of matrices are not the same. Symbolically, if two matrices A and B are equal then write as A=B.

Types of matrices:- 

(i) Column Matrix:- A matrix is said to be a column matrix if it has only one column.

For ex:- 

            is a column matrix of order 4x1.

In general, the order of the column matrix is mx1.

(ii) Row matrix:- A matrix is said to be a row matrix if it has only one row.

For ex:-

           is a row matrix of order 1x4.

In general, the order of the row matrix is 1xn.

(iii) Square Matrix:- A matrix in which no. of rows is equal to the no. of columns, then this matrix is known as a square matrix.

For ex:- 

           is a square matrix of order 3.

In the square matrix, m=n, then the order of the matrix is m.

NOTE:- In the square matrix, elements a11,a22,a33……....ann are said to be a diagonal matrix. For ex:-

            diagonal elements are 0,4,8.

(iv) Diagonal Matrix:- A square matrix is called a diagonal matrix if its all non-diagonal elements are zero.

For ex:-

             are diagonal matrices of order 2 and 3.

(v) scalar matrix:- A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.

For ex:-

           are scalar matrices of order 2 and 3.

(vi) Identity matrix:- A square matrix in which all diagonal elements are 1 and rest are all zero is called identity matrix.

For ex:-

           are identity matrices of order 2 and 3.

⇒ All identity matrices are scalar matrices but all scalar matrices are not identity matrices.

(vii) Zero Matrix:- A matrix is said to be zero matrix or null matrix if all elements are zero. It is denoted by O.

For ex:-

         are zero matrices of order 2 and 3.

Transpose Matrix:- Transpose of matrix A is obtained by interchanging the rows and columns of matrix A. Transpose of the matrix is denoted by A’ or AT.

If A=[aij]mxn, then AT=[aij]nxm.

Properties of the transpose of the matrices:- For any

matrices A and B  

(i) (A’)’ = A

(ii) (kA)’ = kA’ (Here k is any constant)

(iii) (A + B)’ = A’ + B’

(iv) (AB)’ = B’A’

Symmetric Matrix:- A square matrix A is said to be symmetric if the transpose of A is equal to A that is A’ = A.

For ex:-

           

Skew Symmetric Matrix:- A square matrix A is said to be skew-symmetric matrix if transpose of A is equal to the negative of A that is A’ = -A

For ex:-

         

NOTE:- Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Let A be a square matrix then 

               Here (A+A’) is a symmetric matrix and (A-A’) is a skew-symmetric matrix.

     Proof:- Let B=A+A’............(1)

     ⇒ B’ = (A+A’)’

     We know (A+B)’ = A’+B’

     ⇒ B’ = A’+(A’)’

     We know (A’)’=A

     ⇒ B’ = A’+A.........(2)

     From Eq. 1 and 2:-

     ⇒ B = B’

    So B is a symmetric matrix then (A+A’) is also a symmetric matrix.

     Let C=A-A’.........(3)

     ⇒ C’ = (A-A’)’

     We know (A-B)’ = A’-B’

     ⇒ C’ = A’-(A’)’

     We know (A’)’=A

     ⇒ C’ = A’-A

     ⇒ C’ = -(A-A’)

     From Eq. 3:-

     ⇒ C’ = -C

     So C is a skew-symmetric matrix then (A-A’) is also a skew-symmetric matrix.

     Thus any square matrix can be expressed as the sum of symmetric and skew-

     symmetric matrix.

Operations on matrices:- There are certain operations on matrices, namely, the addition of matrices, multiplication of a matrix by a scalar, difference, and multiplication of matrices.

(1)Addition of matrices:- Addition of two matrices is a matrix obtained by adding the corresponding elements of the given matrices. For the addition of matrices, it is necessary that both the given matrices are of the same order.

For ex:- 

                     

Properties of matrix addition:- The addition matrices satisfy the following properties:

(i) Commutative Law:- If A and B are two matrices of the same order then A+B=B+A.

           Proof:- Let A = [aij] and B = [bij] are of same order

mxn

           Then A+B = [aij]+[bij] = [aij+bij] = [bij+aij]

           ⇒ A+B = [bij]+[aij] = B+A (Hence proved....)

(ii) Associative Law:- For any three matrices A,B and C associative law is (A+B)+C = A+(B+C)

           Proof:- Let A = [aij] , B = [bij] and C = [cij] are of same

order mxn

           Then (A+B)+C = ([aij]+[bij])+[cij] = ([aij+bij])+[cij] =

[aij]+[bij]+[cij]

           We know addition is associative(i.e 1+(2+3) =

(1+2)+3  )

           ⇒ (A+B)+C = [aij]+([bij]+[cij]) = A+(B+C) (Hence

proved....)

(iii) Existence of additive identity:- If A be an mxn matrix and O be an mxn zero matrix, then A+O=O+A=A. So O is called the additive identity of matrix addition.

(iv) The existence of additive inverse:- If A be any matrix and another matrix is -A such that A+(-A)=O=(-A)+A. So -A is the additive inverse of A  or negative of A.

NOTE:- If the order of A and B are not the same then A+B is not defined.

(2) Multiplication of a matrix by a scalar:- When a given matrix is multiplied by a scalar then every element of the matrix is multiplied by that scalar.

For ex:-

          

Negative of a matrix:- The negative of a matrix is denoted by -A.(i.e. -A = (-1)A )

For ex:-

            

Properties of scalar multiplication of a matrix:- If A=[aij] and B=[bij] be two matrix of the same order; m and k are scalars, then

(i) k(A+B) = kA+kB

Proof:- k(A+B)=k([aij]+[bij])=(k[aij]+k[bij])

⇒ k(A+B) = kA+kB (Hence proved....)

(ii) (m+k)A = mA+kA

Proof:- (m+k)A = (m+k)[aij] = m[aij]+k[aij]

⇒ (m+k)A = mA+kA (Hence proved....)

(3) Difference of matrices:- If A and B are two matrices of the same order then the difference of matrices is the sum of matrix A and matrix (-B).

For ex:- 

           

(4) Multiplication of matrices:- If A and B two matrices then multiplication of matrices is possible when no. of columns of A is equal to the no. of rows of B. For multiplication of matrices, multiply rows of A and columns of B, multiply them element-wise and then take the sum of them. 

For ex:-

         

NOTE:- If the order of matrix A is mxn and the order of matrix B is nxp, then multiplication is possible and the order of AxB is mxp.

NOTE:- If AB is defined then it is not necessary that BA is also defined. But if A and B both are square matrices of the same order then AB and BA are defined.

Properties of multiplication of matrices:- 

(i) Non-commutativity:- If AB and BA are both defined, it is not necessary that AB=BA.

(ii) The associative law:- For any three matrices A, B, and C, then(AB)C=A(BC) whenever both sides of the equality are defined.

(iii) The distributive law:- For three matrices A, B, and C, then 

(a) A(B+C) = AB+AC

(b) (A+B)C = AC+BC, whenever both sides of the equality are defined.

        (iv) The existence of multiplicative identity:- For

every square matrix A, there 

         exist an identity matrix of the same order such that

IA=AI=A.

Invertible matrices:- If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB=BA=I, then B is called the inverse matrix of A and it is denoted by A-1. Thus A is said to be invertible.

NOTE:- If B is the inverse of A, then A is also the inverse of B.

For ex:-

             

NOTE:- If A and B are two invertible matrices of the same order, then (AB)-1 = B-1A-1.

Proof:- From the definition of the inverse matrix, We know that AA-1=I (Here I is an Identity matrix)

So the inverse of AB is (AB)-1

then (AB)(AB)-1=I

Premultiply both the sides by A-1 :-

⇒A-1(AB)(AB)-1=A-1I

We know A-1I = A-1

⇒(A-1A)B(AB)-1 = A-1

We know A-1A=I and IB(AB)-1=B(AB)-1

⇒B(AB)-1=A-1

Premultiply both the sides by B-1 :-

⇒B-1B(AB)-1=B-1 A-1

We know B-1B=I and I(AB)-1=(AB)-1

⇒(AB)-1=B-1 A-1  (Hence proved....)

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