Combinations problems and word problems
Question 1 - Question: Answer: As n represents all positive integers, we have Multiplying the above terms of both sides respectively, we get Multiplying both sides of inequality by n!, we g..
Question 1 - Question: Answer: As n represents all positive integers, we have Multiplying the above terms of both sides respectively, we get Multiplying both sides of inequality by n!, we g..Question 7
Question: Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Find the number of words which have atleast one letter repeated. Answer: Total number of 5 letter words from given 10 letters = 10 5 (if ..
Matrices and Determinants Summary
. If A =[a i j ] m x n is a matrix of order mxn. The minor of a i j of |A|, denoted by M i j , is given by the determinant which is obtained by deleting i t h row j t h column of |A|. The co-factor of the determinant of the A = [a i j ] m x n , denoted by A i j is given by A i j = (-1) i + j M i j ..
. If A =[a i j ] m x n is a matrix of order mxn. The minor of a i j of |A|, denoted by M i j , is given by the determinant which is obtained by deleting i t h row j t h column of |A|. The co-factor of the determinant of the A = [a i j ] m x n , denoted by A i j is given by A i j = (-1) i + j M i j ..Properties of Inverse of Matrix
In other words, a square matrix A is invertible if and only if A is a non-singular matrix. (c) If A and B are invertible square matrices, then (AB) - 1 = B - 1 A - 1 (d) If A and B are two non-singular square matrices of the same order, then AB and BA are also non-singular matr..
In other words, a square matrix A is invertible if and only if A is a non-singular matrix. (c) If A and B are invertible square matrices, then (AB) - 1 = B - 1 A - 1 (d) If A and B are two non-singular square matrices of the same order, then AB and BA are also non-singular matr..Definition
Let R i denotes the i t h row of the matrix A = [a i j ] then the elementary row operations on the matrix A are defined as: 3. R i g R i + kR j means multiply each element of j t h row by k and add it to the corresponding elements of i t h row. The corresponding column tra..
Let R i denotes the i t h row of the matrix A = [a i j ] then the elementary row operations on the matrix A are defined as: 3. R i g R i + kR j means multiply each element of j t h row by k and add it to the corresponding elements of i t h row. The corresponding column tra..Definition of a Matrix
A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital lette..
Introduction
A set of numbers arranged in a definite order according to some definite rule is called a sequenc..
Note 2:
From the definition, it is clear that if B is the inverse of A, then A is the inverse of ..
Sequences Introduction
Introduction - A set of numbers arranged in a definite order according to some definite rule is called a sequence . Sequences have wide applications. For example, the amount of money in a fixed deposit in a bank, over a number of years increases in a sequenc..
Note:
If two matrices A and B are not of same order, A + B and A - B are not defined..
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