Let a and b belong to Sn. Prove that ßαß-1 and a are both even orboth odd.

Q: For A ∈ Mn(C) show that A = A1 + iA2 uniquely with A1 and A2 selfadjoint. Show that A is normal if…

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Q: Let n, k, m EN with n2k21. prove that

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Q: Let A mabriix show that any two of the properties a ATA A'A-T A-1 imply the other.

A: The square matrix are easier to obtain products and follow many rule of matrix multiplications and…

Q: ands 1. Let m*(A) = 0. Prove that for every B, m" (BUA) = m" (B). %3D %3D

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: If sup(S) exists, then it is unique ylgn ihi

A: The solution is given as

Q: If T is self-adjoint, then S*TS is self adjoint for all S; if S is invertible and S*TS is…

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Q: If B is invertible, prove that det(B-'AB) = det(A).

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Q: 1. Prove that the followingn is a theorem in SD. Av - A

A: Theorem :-

Q: Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q.…

A: We will give two proofs one based on partial sums and the other will be a direct proof.

Q: A The legih in pnefers of the fout sider f a building angx Cox- 5-9*, and Axt a. Calculate The…

A: Solution :   (a)   Total length around the building               = 3x + (6x - 5) + 9x + (4x + 1)…

Q: Let A = d e Assume that det (A) = -13, compute the following.

A: Properties of Determinants :If we given A , nxn matrix then i) The value of det(cA) = cn det(A)…

Q: 2. Let A € M₂ (R) be given. (a) If AA¹ = 0, prove that A = On. Does this hold if A € M₁ (C)? (b)…

A: This is a problem of Linear Algebra.

Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

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Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

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Q: By using caushy Rieman prove that ·C. Prove thatd az az e : ae

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Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: b) If d = 1, prove that (1 if a is odd gcd(a, 2b) %3D if a is even

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Q: Prove that if AT A = A, then A is symmetric and A = A².

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Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: - Prove the following: gcd(m,n) = 1 + 3x, y E Z such that mx + ny = 1. One direction can be done by…

A: We will prove this by using some properties of division. One way will be done with the help of…

Q: Let j = 7 and U=span(ü), with ü = -6. Write j = ã1+ã2, where 1 in U and az in U-.

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Q: If E, and E2 are measurable then (E,UE,) is not measurable .

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Q: 2. Let R = {(x, y) : x, y E R}. Prove or disprove that R is reflexive, symmetric and/or transitive…

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Q: Given an n x n-matriz A over a endent iff det(A) = det(A¹,...,A") =

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Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

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Q: For allne z, gcdin,nt)=l.

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Q: If the i, j entry of A is i times j, show that det A = 0. (Exception when A = [ 1 ] .)

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Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Let a and b be any real numbers. Prove that Cov (ax, by) = abCov (x, y).

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Q: Prove that if Tr(AT A) = 0, then A = O .

A: Proof:- Suppose A is a square matrix of order n, That is, A=aij For 1≤i,j≤n Now we need to prove…

Q: Prove that, IXI<a Rf and If -aL X Z a

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

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

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Q: Let gilx) i for n z2 i define gncr) -gn1(g,(), = 2ー× Deter mine the Intervai Such that g2avu CA) is…

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Q: Let A E Mnxn (F) and B = aA, with a E F. Prove by induction that det(B) all positive integer n. a"…

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Q: Prove that there exist c1, c2 ER such that for all ne N C1 E1 sin (i)<c2 =D1

A: We will use the range of the sine function to determine the real constants c1 and c2

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Ands 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B).

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Q: prove that tr(aA+bB)=atr(A)+btr(B)for any A,B belongs to Mmxn(F)

A: Let A=aij and traceA=trA=∑i=1maii.. Note that, aA+bB=aaij+bbij. Then,…

Q: Do the Same cardimalityq Eitherr give an between them or show no Such bijcction exists- set [o1] and…

A: It is given that

Q: Prove for all o, TE Sn that oro-1 € An. TTO

A: Permutation

Q: When A has full rank, the pseudo-inver en m≥n, and as A+ = AT (AAT)-¹ when n A = I, so A+ is a left…

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Q: Let fons =loag, Cln). Find t'ot at

A: Given, f(x)=logx(ln(x))

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

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Q: Apply any method to prove that alb (or a nand b), which is defined to be ¬(a ^ b) is complete. Write…

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Q: Which theorem can be used to prove that ΔROM ΔΡΜΟ M P.

A: We can answer the question as below

Q: Prove that for all k E N, k ose [coske -sink0j cose Lsinke cosko

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Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below

Q: Prove that det(kA) = kn det(A) for any A ∈ Mn×n(F).

A: In determinant the scalar quantity is taken common from a single row or a column.