Find the inverse of the matrix:

using LU decomposition method with u₁₁ = u₂₂ = u₃₃ = 1.

Calculate the nth divided difference of 1/x, on the nodal points X₀ , X₁ , ......., X_n .

 Using Newton-Rapshson method, find an iterative formula to compute the reciprocal of a natural number N.

 Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:

i) The equation x³ 4x 16 = 0 has a root in the interval [3, 4].

ii) The order of convergence of the secant method is 0.62.

iii) For the system of linear equations:

5x + y + 2z = 34

4y − 3z = 12

10x − 2y + z = −4

the matrix is diagonally dominant.

iv) The numerical method:

is relatively stable.

 v) The method:

converges to 1.5 for any choice of initial approximation.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.

(a) Every subgroup of S₃ is normal.

(b) Every abelian group is cyclic.

(c) In a ring with unity, the sum of any two units is a unit.

(d) If a field has characteristic p, p a prime, the field is finite.

(e) If every element in group has finite order, the group is finite.

 Let R be a ring (not necessarily commutative) and I and J be ideal of R. Show that I ∩ J and I + J = {a + bla ∈ I, b ∈ J} are ideals of R.

Check whether or not (3) is a maximal ideal in Z₉.

Show that (x, 5) is not a principal ideal in Z[x].

 Let S¹ = {z ∈ C | |z| = 1} and Un = {z ∈ C* | z" = 1} for n ∈ N. Check that U_n ⊆ S¹. Further, show that U_n ≤ S¹.

Let F be a field and let f(x) ∈ F[x] be irreducible in F[x]. Show that the ideal (f(x)) is a maximal ideal in F[x]. Use this to deduce that Q[x]/(x² + 6x3 +12) is a field.

Find the order of each of the elements in U(15). Is U (15) cyclic? Justify your answer.

If H and K are normal abelian subgroups of a group, and if H∩ K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.

Find the gcd of the polynomials x4 + 3x + 2 and x³ + 3x² + 5x + 3.

 Let   . Check whether Check whether R is a subring of M₂(R). Is R an ideal of M₂(R)? Justify your answer.

Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.

Calculate the following:

(i) (3x² + 4x + 1) + (3x3 + 4x² + 2x + 3) in Z₅[x].

(ii) (3x²+2x+6). (3x3+4x+5) in Z₇[x].

Let R be a ring in which a² = a for all a ∈ R. Show that a = -a and R is commutative.

Let R = Z₂₀

If F is a field, show that U(F[x]) = F.

Let a = (125), β = (1432) ∈ Ss. Compute σ = α β-¹. Write as a product of transpositons. What is the signature of σ?