Prove that for all values of the constant A, eAInt = e In.

Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

Q: Prove that if Xn X and Ync, then (Xn, Yn) → (X, c).

A: Please see the answer innext step

Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: let theorem a, bor about a) Find such that acb. Use the axioms and that : prove R to b) Find CEIN:…

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Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: 25 If A E M, is symmetric, prove that AT is symmetric.

A: Symmetric matrix

Q: Find all values of a E R such that Q is positive definite, and all values of a E R such that Q is…

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Q: The ODE M(z. v) dz + N(z. v) du = 0 is exact if and only if

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Q: For z C, show that the solutions of cosh(z) = 1 are z= i2km, for ke Z.

A: We need to prove that the solutions of the equation: cosh(z)=1 are z=i2kπ, where z∈ℂ, k∈ℤ.

Q: find the values of c that satisfy Rolle's Theorem

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Q: .Use the De Moivre's Formula to show that if z = (cose + isine), then z" + ; ÷= 2cos (n8) and (금) 3…

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Q: Prove that TIN) F(a – p) O< Re

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Q: Suppose that C> 1. Prove that if t > Vm ln C+In C, then ()/F) >C.

A: We will use the following Strining's approximation :  n! ≈ √(2πn) e-n nn To prove the given…

Q: Show that if x ~ Nm (u, 2), where N is positive definite, then (x – u)'N-I (x – µ) ~ Xm: 2

A: Given  x~Nmμ,Ω, where Ω is positive definite.

Q: Let x e R,-1. Prove that VNEN (1+x)" >1+ nx

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Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: Prove that Z(Sn) = {e} for n ≥ 3.

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Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: Let Be M, such that the entries bij are constants. Prove that tr(ek 3k; det(I – tB) = e

A: It is given that B∈Mn entries of bij are the constant. Let bij=e Therefore, the standard matrix is,…

Q: Prove that eie ei(0+2K) ,where k = 0,±1, +2,...., = D

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Q: 1. ( Prove that for any x, y e R, we have ||| - ||y|| < |× -y|l. 2. Let u e R. Prove that 0 u 0,…

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Q: If A is similar to B, prove that AT is similar to BT.

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Q: If e is the unity in an integral domain D, prove that ( - e )a = - a for all a ϵ D.

A: Given that e is the unity of an integral domain D. Need to prove that (-e) a = -a.

Q: prove that T= S.

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Q: F(p), then prove that

A: simple use of integral transformation

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Find e² in the form of u + jv and le²| for z = 2 + 3nj

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Q: (a) Let r = ri+ yj + zk and let r = ||r||. Prove that - (1) a

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Q: Let a = 8 and b = 9. Find m,n E Z such that am+bn =1.

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Q: Let BE Cnxn. If z" Bz = 0 for all z E C, then show that B = 0.

A: Given expression is

Q: (1) In the following sketch, prove that Â, +Ĉ, = B.

A: Given: The figure To prove: A^1+C^1=B^.

Q: 1. Prove that L(t)=. s2

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Q: Let r E Rt. Prove that if z – 2x2 +2 2021. -

A: Let x2=t, Now if x∈R+⇒t∈R+ ⇒t2-2t+2≤0⇒t-12+1≤0⇒t-12≤-1which is not possible for any value of t, for…

Q: Find the value(s) of k such that A is singular.

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Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: Let z E C. Prove that if 1+z +...+z"-1 – nz" = 0, then |2| <1.

A: Given that, z∈C and, 1+z+...+zn-1-nzn=0 It is known that, 1+z+z2+z3+....+zn-1=1(rn-2-1)(r-1) So the…

Q: Please solve...

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Q: Let A e Mnxn(F) be upper triangular. Prove that det(A) is the product of the diagonal entries.

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Q: Let E/F be Galois and f e F[r] irreducible. Prove that any two irre- ducible factors of ƒ € E[x]…

A: Given : E | F is a Galois extension and f ∈ F [x]  is irreducible over F .  To prove : Any two…

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: Prove that (A+B)T=AT+BT

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Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: 2. Give the condition which ensure that |ez| < 1 where z in C.

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Q: Show that Oa = {a} if and only if a e Z(G).

A: For a∈G, Oa denote the orbit of 'a' under the conjuction action.The orbits under the conjucation…

Q: Prove that e“ p.w. on R. n! n=0 M8

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Q: Prove that (A')' = A for each A E Mmxn(F).

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