Class 12 Chapter 3 (Matrices) class Notes

Chapter - 3

MATRICES

Applications 

The below are the uses of matrices: 

•  Compact and simple methods of solving system of linear equations. 
•  Used as a representation of the coefficients in system of linear equations .
•  Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analysing the results of an experiment etc.
•  Used in physical operations such as magnification, rotation and reflection through a plane .
•  Used in cryptography. 
•  Also used in various branches of sciences like genetics, economics, sociology, modern psychology and industrial management .
•  Used in 3D math, where they are primarily used to describe the relationship between two coordinate spaces.

Matrix 

Definition: 

A matrix is an ordered rectangular array of numbers (may be real or complex) or functions. 

 Elements or Entries of the matrix: The numbers or functions in the array 

 Rows of Matrix: The horizontal lines of elements 

 Columns of Matrix: The vertical lines of elements 

 General Format:

      Simple Matrix: 

                  • Has one Row and Column

                  • Represented as (Row, Column) 

     Higher Level Matrix 

                • Has many Rows and Columns 

                • Represented as below:

              • Furthermore, Generalized format is as below:

           (Or) 
           A= [aij]m×n
           Where, 1 ≤ i ≤ m, 1 ≤ j ≤ n and i, j € N

           • ith Row Elements are aᵢ₁, aᵢ₂, aᵢ₃,......... aᵢₙ

           • jth  Row Elements are a₁j, a2j, a3j,......... anj

 Order of a Matrix 

  If a matrix has "m" rows and "n" columns, then order of the matrix is m × n .

The total numbers of elements in the matrix of m × n will be mn 

  Eg: Order of the below matrix is 3 × 2

           With total number of elements = product of 3 and 2 = 6 

 Any point (x,y) in a plane can be represented in the matrix format as below:

 Suppose if the vertices of a Quadrilateral ABCD are given then it can be represented as matrix format as below 

Points: A(1,0), B(3,2), C(1,3) and D(-1,-2)

 Suppose if the matrix has 8 elements then all possible order of the matrix can be found as below.

            ● Find the Factors of 8 → 1, 2, 4, 8 

            ● Pair up them in all possible ways → (1,8), (2, 4), (8,1), (4, 2) 

 Types of a Matrix :-

  1. Column Matrix 

 A matrix having only one column and any number of rows

 General Form : A= [aij]m×1, order of the matrix is m×1

 2. Row Matrix

 A matrix having only one row and any number of columns

 General Form = A= [aij]1 x n, Order of the matrix is 1 × n 

 3. Square Matrix 

  A matrix of order m × n, such that m = n

 General Form = A = [aij]m×m, Order of the matrix is m 

 4. Diagonal Matrix

  A square matrix is said to be a diagonal matrix if all its non-diagonal elements are zero

 General Form = A = [bij]m×m, Order of the matrix is m where bij=0, if i≠j 

 5. Scalar Matrix 

 A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal

 General Form = A = [bij]m×m, Order of the matrix is m where 
         ● bij, =0, if I≠j 
         ● bij =k, if i = j for k = constant 

 6. Identity Matrix 

 A square matrix in which elements in the diagonal are all 1 and rest are all zero

 General Form A= [bij]m×m, Order of the matrix is m where
         ● bij, =0, if i≠j 
         ● bij =1, if i = j 

 7. Zero Matrix 

 A matrix is said to be zero matrix or null matrix if all its elements are zero

 Denoted by O

 8. Equal Matrices 

 Two matrices are said to be equal if 
               ●  they are of the same order 
             ●  each element of A is equal to the corresponding element of B

 Using the concept of Equal matrices, we can find the unknown values of any matrix.

We can equate the corresponding terms from the given matrices: 

x + 3 =0                     z+4 = 6             2y - 7 = 3y - 2
a -1= - 3                     0 = 2c + 2         b- 3 = 2b + 4,  

Simplifying, we get
 a = - 2, b = - 7, c = 1, x = - 3, y = -5, z = 2

 Operation on Matrices 

 Addition of Matrices 

 Let A and B be two matrices each of order m × n. Then, he sum of matrices A + B is defined only if matrices A and B are of same order. 
            ● If A = [aij]m×n and B = [bij]m×n 
               Then, A + B = [aij + bij]m×n

  General format

Example

  When adding two matix A & B, if the order is not same then A + B is not defined 

Properties of Addition of Matrices

  If A = [aij], B = [bij] and C = [cij]are three matrices of order m × n, then

        Commutative Law 
                   A + B = B + A 

        Associative Law 
                 (A + B) + C = A+ (B + C) 

        Existence of Additive Identity 
                   A zero matrix (0) of order m × n (same as of A), is additive identity, if A + 0 = A = 0 + A 

        Existence of Additive Inverse 
                   If A is a square matrix, then the matrix (- A) is called additive inverse, if A +(- A) = 0 = (- A) + A 
                      • - A is the additive inverse of A or negative of A. 

      Cancellation Law 
              A+B = A + C = B = C (left cancellation law)
              B+A = C + A = B = C (right cancellation law)

 Subtraction of Matrices

  Let A and B be two matrices of the same order, then subtraction of matrices, A - B, is defined as A -B = [aij + bij]n×n, where A = [aij]m× n, B = [bij]m×n

Here Order of A is 3 × 2 and Order of B is 3 × 2 therefore, subtraction is possible

font-size: large;"> Multiplication of of Matrices 

Multiplication of a Matrix by a Scalar
Let A = [aij]m×n be a matrix and k be any scalar. Then, the matrix obtained by multiplying each element of A by k is called the scalar multiple of A by k and is denoted by kA, given as 
kA=[kaij]m×n

Properties of Scalar Multiplication If A and B are matrices of order m×n, then 

         •  k(A + B) = kA + kB 
         • (kı + k2)A = k1A + k2A 
         • kık2A = k1(k2A) = k2(k1A) 
         • (- k)A = - (kA) = k( - A) also called as negative of a matrix

 Multiplication of two Matrices 

Consider two matries A and B, then for the multiplication to e possible umber of columns in A should be equal to the number of rows in B. 

If A = [aij] of order m×n and B = [bij] of order n×p then the product C = [cij] will be a patrix of the oder m × p

Notes:-

  If AB is defined, then BA need not be defined. 
If A, B are, respectively m × n, k × I matrices, then both AB and BA are defined if and only if n = k and 1 = m.
 If both A and B are square matrices of the same order, then both AB and BA are defined.
If AB and BA are both defined, it is not necessary that AB = BA. 
If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix

 Properties of Multiplication of Matrices

 If A = [aij], B = [bij]and C= [cij]are three matrices of order m × n, then

Commutative Law 
        • AB ≠ BA 
Associative Law 
        • (A B)C = A(BC) 
Distributive Law 
        • A (B+C) = AB + AC 
        • (A+B) C = AC + BC, whenever both sides of equality are defined. 
  Existence of Multiplicative Identity 
        •  For every square matrix A, there exist an identity matrix of same order such that IA = AI = A 
Cancellation Law 
        • If A is non-singular matrix, then .
            • AB = AC = B = C (Left cancellation law) 
            • BA = CA =B = C (Right cancellation law) 
            • AB = 0, does not necessarily imply that A = 0 or B = 0 or both A and B =

 Transpose of a Matrix 

Let A = [aij]m×n be a matrix of order m × n. Then, the n×m matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by 'or AT.

  A' = AT = [aij]n×m

 Properties of Transpose 

(A')' = A 
(A + B)' = A' + B' 
(AB)' - B'A' 
(kA)' - kA' 
(AN)' = (A')N 
(ABC)' =C' B' A' 

Symmetric & Skew Symmetric Matrices 

A square matrix A = [aij] is said to be symmetric if A'= A

Note :-

  If A is Symmetric then, [aij] = [aji]
   If A is Skew Symmetric then, [aij] = -[aij]
All the diagonal elements of a skew symmetric matrix are zero. 

Theorem 1 

For any square matrix A with real number entries, A + A' is a symmetric matrix and A- A' is a skew symmetric matrix. 

Proof 
      Let B = A + A', then 
      B' = (A + A')'

       = A' + (A')' (as (A + B)' = A' + B') 

       = A' + A (as (A')' = A) 
       = A+ A' (as A + B = B + A) 
       = B
       Therefore B = A + A' is a symmetric matrix 
       Now let C = A - A'
       C' = (A - A')' = A' - (A)' (Why?) 
       = A' - A (Why?) 
       =- (A - A') = - C 
       Therefore C = A - A' is a skew symmetric matrix 

Theorem 2

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

Proof 
      Let A be a square matrix, then we can write

 Elementary Operation (Transformation) of a Matrix 

The six operations (transformations) on a matrix, three of which are due to rows and three due to columns

 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. In that case A is said to be invertible

Note :-

A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. 

If B is the inverse of A, then A is also the inverse of B

Inverse of a Square Matrix 

Let A be a square matrix of order n, then a square matrix B, such that AB = BA = I, is called inverse of A, denoted by A⁻¹ 
AA⁻¹ - A⁻¹A =1 

Theorem 3 (Uniqueness of inverse) 

Inverse of a square matrix, if it exists, is unique. 

     i.e Let A = [aij] be a square matrix of order m and B and C be two inverses of A. 
    To prove : B- C 

Proof :-

      Since B is the inverse of A 
      
      AB = BA = I.......(1) 
      Since C is also the inverse of A 
      AC = CA = I.......(2) 
      Thus B = BI = B (AC) = (BA) C = IC = C 

Theorem 4

If A and B are invertible matrices of the same order, then (AB)⁻¹ =B⁻¹A⁻¹. 

Proof : 

From the definition of inverse of a matrix, we have 
(AB) (AB)⁻¹= 1 
A⁻¹ (AB) (AB)⁻¹ = A⁻¹I (Pre multiplying both sides by A⁻¹) 

= (A⁻¹ A) B (AB)⁻¹ = A -1 (Since A⁻¹ I = A⁻¹) 

= IB (AB)⁻¹ = A⁻¹ 
= B (AB)⁻¹ = A⁻¹ 
= B⁻¹ B (AB)⁻¹ = B⁻¹ A⁻¹ 
= I (AB)⁻¹ = B⁻¹ A⁻¹ 
Hence, (AB)⁻¹ = B⁻¹ A⁻¹

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