Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Chapter 3 Matrix Class XII : Basic Concepts

Class 12 Chapter 3 Matrices
Complete explanation, basic concepts and formulas based on matrices. Matrix method of arranging the terms , addition, subtraction, multiplication , and transpose of the matrices.

Solution of Important questions of NCERT Chapter 3 Matrices

Basic Concepts Based on Chapter 3 (Matrix) Class 12

Definition :  A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

Order of a Matrix:
A matrix having m rows and n columns is called a matrix of order m x n
In general a matrix of order m x n  can be arranged in a rectangular array as
\[\begin{bmatrix} a_{11} & a_{12} & a_{13} &...... & a_{1n} \\ a_{21} & a_{22} & a_{23} &...... & a_{2n} \\ a_{31} &a_{32} & a_{33} &...... & a_{3n} \\ ....& .... & ... & ...... & .... \\ ....& .... & ... & ...... & .... \\ a_{m1}&a_{m2} &a_{m3} & ...... &a_{mn} \end{bmatrix}_{m\times n}\]
a11 means an element in the first row and first column, a23 means an element in the second row and in the third column and so on.

In short a matrix can be written as  A = [aij]m x n , where aij  represent the elements of the matrix, m is the number rows and n is the number of columns.

Types of Matrices

Column Matrix :

If the matrix has only one column then it is called column matrix.
\[\begin{bmatrix} a\\b \\c \end{bmatrix}\]
Row Matrix : If a matrix  has only one row then it is called a row matrix.
\[\begin{bmatrix} a & b &c \end{bmatrix}\]
Square Matrix : A matrix in which number of rows and columns are equal is called a square matrix.  For Example:    
\[\begin{bmatrix} a &b \\c &d \end{bmatrix}_{2\times 2}\; \; \; or\; \; \; \begin{bmatrix} a &b &c \\d & e &f \\ 1& 2 &3 \end{bmatrix}_{3\times 3}\]

Diagonal Matrix : A square matrix is said to be diagonal matrix if its all non diagonal elements are zero. For example : 
\[\begin{bmatrix} a &0 &0 \\ 0 &b &0 \\ 0 & 0 & c \end{bmatrix}\]

Scalar Matrix : A diagonal matrix is said to be scalar matrix if its all diagonal elements are equal.  For Example : 
\[\begin{bmatrix} 5 &0 &0 \\ 0 &5 &0 \\ 0 & 0 & 5 \end{bmatrix}\]

Identity Matrix : A diagonal matrix is said to be identity matrix if its all diagonal elements are 1. Example : 
\[\begin{bmatrix} 1 &0 &0 \\ 0 &1 &0 \\ 0 & 0 & 1 \end{bmatrix}\]

Zero Matrix : 
A square matrix is said to be zero matrix if its all elements are zero.
\[O=\begin{bmatrix} 0 &0 &0 \\ 0 &0 &0 \\ 0 & 0 & 0 \end{bmatrix}\]
Equal Matrices : Two matrices A and B are said to be equal if
a) Matrices A and B  are of same order  
b) Corresponding elements of matrix A and B are equal.

Addition of two matrices :

Addition of two matrices is possible if they are of same order

Addition of two matrices is done by adding their corresponding elements.
  
Example : \[\begin{bmatrix} a &b &c \\ x& y & z\\ p & q & r \end{bmatrix}+\begin{bmatrix} 1&2 &3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}=\begin{bmatrix} a+1&b+2 &c+3 \\ x+4 & y+5 & z+6\\ p+7 & q+8 & r+9 \end{bmatrix}\]

Addition of two matrices is not possible if they are not of same order.

Multiplication of a matrix with a scalar

Let A is a matrix and k is a scalar, then multiplication of scalar k with matrix A is obtained by multiplying each element of matrix A by a scalar k.\[If\; matrix\; A=\begin{bmatrix} x &y &z \\ p & q &r \\ a&b &c \end{bmatrix} \; and \; scalar\; k=2,\; then\] \[2A=2\begin{bmatrix} x &y &z \\ p & q &r \\ a&b &c \end{bmatrix}=\begin{bmatrix} 2x &2y &2z \\ 2p & 2q &2r \\ 2a & 2b &2c \end{bmatrix}\]
Properties of matrix addition

If A and B are two matrices then

Commutativity

Matrix addition holds the commutative law :  A + B  = B + A

Associative law

Matrix addition holds the associative law  :  (A + B) + C = A + (B + C)

Existence of Identity : If zero matrix O is added to any matrix A then matrix A is remain unchanged.  A + O = O + A = A ⇒ O is the identity matrix in Matrix addition.

Existence of additive inverse: If A is a matrix then  A + (-A) = (-A) + A = O, then -A is said to be the Additive  inverse of matrix A.

Multiplication of matrices

Product of two matrices  A  and  B is defined if number of columns of first matrix  A is equal to the number of rows of matrix B
Example : If A and B are two matrices such that:  
\[A=\left [ a_{ij} \right ]_{m\times n}\; \; and\; \; B=\left [ b_{ij} \right ]_{p\times q}\]Then product AB exists if  n = p and the order of the resulting matrix AB  is m x q

Note : If product AB is defined then product BA need not to be defined
Product AB and BA both are defined if A and B both are the square matrices of same order.
Example : If A and B are two matrices such that : 
\[A = \begin{bmatrix} 2 &4 &-3 \\ 1 &4 & 3 \end{bmatrix}\: \: and\: \: B=\begin{bmatrix} 2 &-3 \\ 4 &5 \\ 1 & 2 \end{bmatrix}\]
Order of matrix A is  2 x 3, and the order of matrix B = 3 x 2. Number of columns of first matrix is equal to the number of rows of second matrix. 
Therefore product of A and B is defined. And the order of the resulting matrix become  2 x 2
\[AB=\begin{bmatrix} 2\times 2+4\times4+(-3)\times1 &\; \; \; \; \; 2\times-3+4\times5+(-3)\times2 \\ 1\times2+4\times4+3\times1&1\times-3+4\times5+3\times2 \end{bmatrix}\]
\[AB=\begin{bmatrix} 4+16-3 &\; \; -6+20-6 \\ 2+16+3 &\; \; -3+20+6 \end{bmatrix}\]
\[AB=\begin{bmatrix} 17 &\; \; 8 \\ 21 &\; \; 23 \end{bmatrix}\]

Method of multiplying two matrices of order 2 X 2
Method of multiplying two matrices of order  3 X 3



Properties of matrix multiplication.

i) Matrix multiplication is not commutative.  ⇒ AB ≠ BA

ii) If A and B are the diagonal matrices of the same order then multiplication of the matrices is commutative. In this case we can write   AB = BA

iii)  Matrix multiplication is associative   (AB)C = A(BC)

iv) Matrix multiplication holds the Distributive property

A(B + C) = AB + AC  or  (A+B)C = AC + BC

Existence of multiplicative identity :

For every square matrix there exists an identity matrix I of the same order such that  AI = IA = A,  I is the identity matrix for A.
Example
For a square matrix of 2, I is the identity matrix and is given by \[I=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}=I_{3}\]
For a square matrix of 3, I is the identity matrix and is given by \[I=\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0& 0 &1 \end{bmatrix}=I_{3}\]

Transpose of a matrix

If A is a matrix then transpose of the matrix A is obtained by interchanging the rows and columns . It is denoted by A'
Example
\[If\; A=\begin{bmatrix} 17 & 8\\ 21& 23 \end{bmatrix}\; \; then\; \; A' =\begin{bmatrix} 17 &21 \\ 8& 23 \end{bmatrix}\]
\[If\; \; A=\begin{bmatrix} a &b &c \\ x& y & z\\ p &q &r \end{bmatrix}\; \; then,\; A'=\begin{bmatrix} a & x &p \\ b& y &q \\ c& z &r \end{bmatrix}\]

Properties of the transpose of the matrices

i)   (A')'  =  A,   or   Transpose of the transpose matrix is the matrix itself

ii)    (kA)'  =  kA',  where k is any scalar

iii)   (A + B)' = A' + B'

Symmetric and Skew symmetric matrices

Symmetric Matrix : A matrix A is said to be symmetric if  A' = A

Skew symmetric matrix : A matrix is said to be skew symmetric if A' = -A

Theorem 1:
For a square matrix A with real entries A + A' is always symmetric matrix and A - A' is always skew symmetric matrix.

Proof:   Let A + A' = B,  then

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

⇒ B = A + A' is a symmetric matrix

Let C = A - A' ,  then 

C' = (A - A')' = A' - (A')' = A' - A = - (A - A') = - C

Theorem 2: 
Any square matrix can be expressed as the sum of symmetric and skew symmetric matrix.
Proof:
If A is a matrix  and  A' is a transpose of matrix A, then A + A' is always a symmetric matrix and A - A' is always a skew symmetric matrix  
\[\Rightarrow P=\frac{1}{2}(A+A')\; is\; a\; symmetric\; matrix\]
\[\Rightarrow Q=\frac{1}{2}(A-A')\; is\; a\; skew\; \; symmetric\; matrix\]
\[A=P+Q=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')=A\]
⇒ Every square matrix can be written as the sum of symmetric and skew symmetric matrix

 Elementary operations of a matrix

There are mainly six operations on the matrix. Three of which are due to row and other three of which are due to column.

Here we explain only row operations. Column operations are also same.

i) Interchanging any two rows is denoted by  Ri  Rj

ii) Multiplication of all the members of any row by a scalar k is  is denoted by Ri  →  kRi

iii) The addition of the elements of any row with the corresponding elements of any other row multiplied by any non-zero number is denoted by Ri   Ri + kRj

To find the inverse of the matrix by the method of elementary operations
To find A-1 by elementary operations we can either follow row operations or column operations.
If we follow row operations then we write the equation  A = IA

If we follow column  operations then we write the equation  A = AI

Method of Finding the Inverse of 2 x 2 matrix by Elementary Operations

Step 1: For Row operations we write A = IA

Step 2: Make a11 = 1

Step 3: With the help of a11 = 1 make a21 = 0

Step 4: Now make a22 = 1

Step 5: With the help of a22 = 1 make a12 = 0

Method of Finding the Inverse of 3 x 3 matrix by Elementary Operations

Step 1: For Row operations we write A = IA

Step 2: Make a11 = 1

Step 3: With the help of a11 = 1 make a21 and a31 = 0

Step 4: Now make a22 = 1

Step 5: With the help of a22 = 1 make a12 and a32 = 0

Step 6: Now make a33 = 1

Step 7: With the help of a33 = 1 make a13 and a23 = 0

 Invertible matrices

If A and B are two square matrices of the same order such that AB =BA = I, then B is said to be inverse of A and is denoted by A-1  such that A-1  = B. In this case A is said to be invertible.

In other words :
 If A is invertible then A-1 exists  or If  A-1 exists then A is said to be invertible.

Theorem 3 : Inverse of a matrix if exist then it is unique.

Proof:  
If possible let us suppose that the matrix A has two inverse B and C

B is the inverse of A then  AB = BA = I………… (i)

C is the inverse of A then  AC = CA = I ………(ii) 

From  (i) and (ii) we get

AB = AC  ⇒ B = C               ………….(Cancelling A on both side)

     ⇒ Both the inverse are same.

Theorem 4:
 If A and B are two invertible matrices of the same order then (AB)-1  =  B-1A-1

Proof :
By the definition of inverse we have
           (AB)(AB)-1  =  I
Multiplying on both side by A-1  we get

    A-1(AB) (AB)-1  = A-1 I         (A-1A) B(AB)-1  = A-1
           (IB)(AB)-1  = A-1                        B(AB)-1  = A-1
       (B-1B) (AB)-1 = B-1A-1                      I(AB)-1 = B-1A-1     
               (AB)-1 = B-1A-1


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