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Tutor profile: Bhabesh M.

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Bhabesh M.
Tutor for 2 years
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Questions

Subject: Linear Algebra

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Question:

The 3x3 matrix A = [ 1 0 1], [0 2 -2], [0 0 3]. Find the value of A^29.

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Bhabesh M.
Answer:

The complexity of calculation and the power makes is too difficult to find the answer by manually multiplying matrices . But , Using the following result from diagonalization of matrices we have For any matrix A having distinct eigen values D= P^-1 A P. Where D is a diagonal matrix which has diagonal entries as the Eigen values of A and P is an invertible matrix which has Eigen vectors of A as Column vectors. Here our matrix has Eigen Values as 1,2 and 3 found out my simple calculation using the Characteristic polynomial as -λ^3+6*λ^2-11*λ+6 =0 so D=[1 0 0], [0 2 0], [0 0 3] and as P is Invertible so D= P^-1 A P = > P D P^-1 = A = > A = P D P^-1 = > A^29 = (P D P^-1 ) ^29 = > A^29 = P D^29 P^-1 { As (P D P^-1 ) ^29 = P D P^-1 * P D P^-1 *P D P^-1.............*P D P^-1} and P* P^-1 = I (identity matrix) So, A^29= P{ [1 0 0] , [0 2^29 0], [0 0 3^29] } P^-1. where P=[1 0 0], [0 1 0 ], [1/2 -2 1]

Subject: Set Theory

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Question:

In a group of 60 people, 27 like Mathematics and 42 like Physics and each person likes at least one of the Subjects. How many like both Maths and Physics?

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Bhabesh M.
Answer:

Formula to use (Theory behind the problem): n( A U B ) = n(A) + n(B) - n(A and B) or n( A and B)= n(A) + n(B) - n( A U B) Solution : A= No of people who like Maths B= No of people who like Physics n( A and B)= n(A) + n(B) - n( A U B) n( A and B)= 27+ 42 - 60 n( A and B)= 9 So the number of people who like both maths and physics are 9.

Subject: Algebra

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Question:

Does the ideal <x^2 +3> form a field as a quotient field over Q[x] (rationals)?

Inactive
Bhabesh M.
Answer:

An ideal forms a field as a quotient field over a Ring when it is the Maximal ideal. Here the ideal is generated by the polynomial x^2+3. The ideal here will be a maximal ideal iff the polynomial generating it is irreducible. So, the question boils down to decide whether the polynomial is irreducible over Q or not. Now, lets use Eisenstein's Criterion which states : Suppose we have the following polynomial with integer coefficients. P(x)=a{n}x^{n}+a{n-1}x^{n-1}+......... +a{1}x+a{0} If there exists a prime number p such that the following three conditions all apply: p divides each a{i} for i ≠ n, p does not divide a{n}, and p^2 does not divide a{0}, then P is irreducible over the rational numbers. choosing p=3 does the trick. Since, 3 | 3 =a{0} and p^2=9 doesn't divide 3=a{0} and 3 doesn't divide 1 = a{n}. So , Since the polynomial is irreducible over Q, it forms a maximal ideal over it and the quotient ring generated becomes a field

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