CHAPTER - 3 MATRICES CLASS 12th

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Matrix -: A rectangular array of mn numbers in the form of m horizontal lines

(rows) and n vertical lines (columns), is called a matrix of order m x n.

This array is enclosed by brackets [ ], ( ) or || ||.

Each number in the matrix is known as element of that matrix. Generally a

matrix is denoted by A, B, X…. etc. and its each element is denoted by aij

where aij belongs to the ith row and jth column.

Generally m × n matrix is written as A = [

… … …

] = [aij ]m × n

Eg . A = [

] is a matrix of order 2 × 2.

Types of matrices -:

1. Row Matrix – matrix which has only single row & any number of

columns.

eg. A = [ ]1×3 .

2. Column Matrix – matrix which has only single column & any number of

rows.

3. Square Matrix – A matrix in which number of rows are equal to number

of columns.

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eg. A = [

] is a matrix of order 3 × 3.

4. Null Matrix – A matrix in which all elements are Zero. It is denoted by O.

eg. [

] is a null matrix of order 2 × 2.

5. Diagonal Matrix – If all elements of a matrix (except its diagonal

elements are 0) then it is said to be Diagonal matrix.

Note -: Diagonal Elements Should not be Equal.

eg. [

] is a diagonal matrix and also denoted by A = diag [2, 7, 5 ].

6. Scalar Matrix - If all elements of a matrix (except its diagonal elements

are 0) then it is said to be Scalar matrix.

Note -: Diagonal Elements Should be Equal.

You can also say like this -:

(a) aij = 0, ⩝ i ≠ j

(b) aij = k, ⩝ i = j, where k ≠ 0.

eg. [

] is a scalar matrix.

7. Identity Matrix or Unit Matrix – If all elements of a matrix (except its

diagonal elements are 0) then it is said to be Scalar matrix.

Note -: Diagonal elements should be equal to 1.

You can also say like this -:

(c) aij = 0, ⩝ i ≠ j

(d) aij = 1, ⩝ i = j

eg. [

] is a Identity or Unit matrix.

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Properties of Matrix Addition –

Let A, B and C are three matrices of same order, then

(a) Matrix addition is Commutative

i.e. A + B = B + A

(b) Matrix addition is Associative

i.e. (A + B) + C = A + (B + C)

(c) Additive Identity

i.e. A + O = A = O + A

(d) Additive Inverse

i.e. A + (−A) = O = (−A) + A

Here (−A) is additive inverse of A.

NOTE -: Two matrices can be add or subtract only if they have same orders.

Multiplication of Two Matrices -:

If A = [aij]m × n and B = [bij]n × p are two matrices such that the number of

Columns of A is equal to number of rows of B, then a matrix C =[cij]m × p of

order m p is known as product of matrices A and B.

NOTE -: Matrix Multiplication is only possible if the number of columns of first matrix

is equal to the number of rows of second matrix. and the order of Resultant Matrix is

equal to the number of rows of first matrix and number of columns of second matrix.

Properties of Multiplication of Matrices -:

Let A = [aij]m × n , B = [bij]n × p and C =[cij]m × p are three matrices then,

(a) Generally, matrix multiplication is not commutative.

i.e. AB ≠ BA

(b) Matrix multiplication is Associative

i.e. (AB)C = A(BC)

(c) Distributive Law

i.e. A(B + C) = AB + AC

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(d) If In is an identity matrix of order n × n and Im is an identity matrix

of order m × m and A is a m × n matrix then,

Im A = A and AIn = A

Now, if A is a square matrix of order n × n then,

AIn = In A = A

(e) AB = O does not imply that A = O or B = O or both A & B are O.

Transpose of a Matrix -:

If A = [aij] be an m × n matrix, then the matrix obtained by

interchanging the rows and columns of A is called the transpose of A.

Transpose of the matrix A is denoted by A′ or (AT).

In other words, if A = [aij]m × n, then A′ = [aji]n × m.

eg. [

], then A′ = [

]

Properties of Transpose -:

(a) (A′)′ = A (b) (kA)′ kA′ (where k is any constant)

(c) (A + B)′ A′ + B′ (d) (A B)′ B′ A′ (Reversal Law)

Symmetric Matrix -:

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

i.e. [aij] = [aji] for all possible values of i and j.

eg. A = [

] is a symmetric matrix because A’ = A.

Skew - Symmetric Matrix -:

A square matrix A = [aij] is said to be skew - symmetric if A’ = − A

OR

1. aij = − aij , ⩝ i, j

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2. Each diagonal element should be Zero.

eg. A = [

] is a skew - symmetric matrix because A’ −A.

Elementary Operation of a Matrix -:

Any of these operations on a matrix is called an elementary row (or

column) transformation.

(a) Interchanging any two rows (or columns). This transformation

is indicated by

Ri ⟷ Rj (or Ci ⟷ Cj )

(b) Multiplication of the elements of any row (or columns) by a

non-zero number.

Ri ⟷ kRj (or Ci ⟷ kCj )

(c) Addition of a constant multiple of the elements of any row to

the corresponding element of any other row. This

transformation is indicated as

Ri ⟶ Ri + kRj .

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 –:

1. 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.

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

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