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

Q: Verify that the Galois imaginary is primitive in GF(3, x³ + 2x+1)

A: Given:- Verify that the Galois imaginary is primitive in GF(3,x3+2x+1)

Q: 4 Let A b [² }] Find Conditions on 1 c d a,b,c and d that AX =0 that AX=0 has infinitely many…

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Q: If A is invertable then A is diagonalizable. Select one: O True O False

A: If a matrix A of order nxn is invertiable then it has n disctinct eigen values.  And corresponding…

Q: Prove that if A is invertible and AB = 0, then B= 0. Give a counterexample to show that the result…

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Q: =) Prove: If A is invertible, then det(A-1) = ) Prove: If A is inverible, then adj(A) is inve

A: Introduction: In the case of real numbers, the inverse of any real number a was the number a-1, so…

Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: (b) 1 = -4 - 4- 2.

A: Given that

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Assume that A is partitioned as in figure that P is invertible, and that PR = RP. Prove that detA =…

A: Given that, Matrix A is partitioned as, Also, P is invertible, and that PR = RP Since P is…

Q: Show that if A is invertible and DA = CA, then D = C.

A: Given that, A is invertible and DA=CA.

Q: (a) Prove that if A is invertible and BA = CA, then B=C. (b) Give a counterexample to show that the…

A: Given:         Part(a):Since A is invertible then A−1 exist.Now,BA=CA(BA)A−1=(CA)A−1…

Q: Suppose I + AB is invertible. Show that I + BA is also invertible by showing that (I + BA)¯ = I –…

A: Assume that I+AB is invertible.  Then, I+ABI+AB-1=I ⋯ 1. Use equation (1) and calculate…

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: Consider the D-operator P(D) = mD² + 2kD +¢, where m and c are positive. If -c/m < k< 0, then

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Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: Prove that if A is invertible and BA = CA, then B = C.Give a counterexample to show that the result…

A: Given: A is invertible and BA = CA  To prove: B = C

Q: Let f : A → B and g : B → C be bijections. Prove that gof is invertible and that (g o f)-1 = f-1 o…

A: Problem based on function.

Q: 4. Use mathematical induction to prove that V n E Z* U {0}, [I²-o(2i + 1)! 2 ((n + 1)!)**. n+1

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Q: If (a] € Z/mZ is invertible, is [-a] invertible? If yes, prove it and say what the multiplicative…

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Q: 2. Determine all k e R for which A is diagonalizable. [1 k 1 (a) A [1 0 k (b) A= |0 1 0 0 0 k

A: Solve the following

Q: Suppose f(r) = for real numbers a, b, c with both a and e not zero. न्य (a) Find the domain of f.…

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Q: Let T: R" → R" be defined as T(x₁,x₂,...,xn) = (x₂ −X₁, X3 X₂, ..., x₁ - xn) of T Discuss the…

A: The matrix is invertible if the determinant of the matrix is non-zero. The matrix is diagonalizable…

Q: Prove that if A is invertible and AB = O, then B = O.

A: To prove that if A is invertible and AB=0, then B=0.

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: 3. Suppose that BE Mxn(C) and B = 0. Show that I-B is invertible. Hint: Consider I+ B+ B.

A: Given : B is an n×n complex matrix with  B3 = O , where O is zero matrix. To prove :  I-B is…

Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

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Q: Let A be invertible. Prove that At is invertible and (At)−1 = (A−1)t.

A: Given:Let A be invertible. Prove that AT is invertible and (AT)−1 = (A−1) T.

Q: Show that following MA models are equivalent A: Yt = et – Oet-1 for et ~ WN(0, 1) 1 Yt = et for et ~…

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Q: Find the Wronskian of {e¯*, xe-x, x²e¯x}.

A: Given that, The Wronskian of two differential function f and g is f (g') - g (f'). If f1, f2, ...,…

Q: Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizable

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Q: 2. Let k be a fixed real scalar. (a) Show that L: R? R defined by a ka +b

A: Given, k be a fixed real scalar. (a) To prove that Lk: ℝ2→ℝ2 defined by…

Q: b) For 7 in R", we say that is in the kernel of T if T(x) = 0. If x₁ and 2 are both in the kernel of…

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Q: Let A find on Conditions a,bic and d has infinitely that AX =o many solution . what must there Pere…

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Q: . Suppose that T e L(V) and U and W are invariant under T. Prove that U + W is also invariant under…

A: If U is subspace of V and T(U) is subeset of U then U is invariant under T

Q: If A is invertable then A is diagonalizable. Select one: True O False

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Q: In C[-1,1], with the standard inner product f(x) = e² and f(x) = e ² are orthogonal. %3D O True O…

A: Given- fx=ex and gx=e-x. To Check- If the given functions with the inner product are orthogonal or…

Q: Suppose A and B are invertible then det(A) = det(B) %3D O True O False

A: Consider A and B are invertible

Q: 1) Let F := {B E B(R) | B = –B}.Show that B is a o-field.

A: This is a question from Measure Theory.

Q: that N(ry) = N(x)N(y) for all x and y in 22). 2. If x is invertible in Z[1], show that N(x) must…

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Q: Let A = k 0 1 k - 1 2k k+ 1. Use cofactor expansion to determine which values of k make A…

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Q: Show that if k is a scalar and A is n × n, then det(k A) = kn det(A).

A: Show that if k is a scalar and A is n×n, then det(kA)= kndet(A)

Q: Show that if B is m × n, then BTB is positive semidefinite; and if B is n × n and invertible, then…

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Q: (a) Verify the Cayley-Hamilton theorem. (b)  Use the theorem to express A2 and A3 as linear…

A: Part (a), Let, A=4-112 Now, the characteristic equation of A,…

Q: Prove that if A is invertible and AB=0, then B=0.

A: Given:

Q: Let K = Q(i) denote the field of Gaussian numbers. Which of the following elements of OK are…

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

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

A: ANSWER:

Q: Let A and P be an n x n matricies, where P is invertible. Show that det(PA(P-1))=detA.

A: To show that,

Q: Determine whether the following are linear operatorson Rn×n. L (A) = A − AT

A: