Prove that A=I-N+N²-N³ for A & N definec as follows: N=A-I (identity), where |1 0 0 an & I= 0 1 0. 0 0 1 A=0 1 0 0 1

Q: (a) Show that for all n € Z, n ≥ 1, if A and B₁, B2, ... are any sets, then (c) n i=1 (b) Show that…

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Q: et z = 6ex²y³ find O 12xy³ex²y³ O 18x²y²ex²y³ O 18xy³x²y³ O 12x²y²ex²²³ дz ax

A: Given, z=6ex2y3

Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: Prove that z =z iff z is real.

A: z represents a complex number which is in the form x+iy, where x and y are the real numbers and i is…

Q: Let i = (– 4, 4, 5) and i = (6, – 3, – 2). %3D • Hint

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Q: = a O a = a for all positive integers a. = a O b = b O a for all positive integers a and b. = If a >…

A: We apply the given formulas repeatedly.

Q: Let A= 1 0 1 0 0101 001 1 1100 What is the rank of A?

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Q: 2. Let f: ZxZ-ZxZ be defined by f(m, n) = (3m+n,n²) a. Is fone-to-one? b. Is fonto?

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Q: Let X₁, X₂,..., X be a r.s. from N(,0) and Y₁, Y₂...., be another independent r.s. from m…

A: Solution

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r>d/2 where x, y, z belongs to Rn So locus z…

Q: -1 2 1 Let A Compute AAT and AT A. 1 0 1 1 -1

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Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Prove that if 0 < a < y, then ỹ – VI < Vy– a.

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Q: Let Sy := {1-(-1):NEN. Find iNf Sy and sup Sy

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Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: Let α belong to Sn. Prove that |α| divides n!

A: The symmetric group ​ is the group of permutations on n objects. Its order is .

Q: 14. Prove that AC B if and only if An BC= 0.

A: The objective is to prove the above question.

Q: 1. Let A=(a)xn be symmetric and positive definite. Show that a) a>0, i=1,2,...,n. |a|< (aa)², i,…

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Q: |°z\+|'z| |*z\-|°z| Z1 +z2 Z 2 +|z. Z3 + z, Prove: If Fs|#/z,) show that ' 4 VI

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Q: Show that det [1 1 0 0 1 1 0 0 001 0 0 0 1... 0 0 ⠀⠀⠀ 0000 1000 0 0 ... 1 1 01 = 1+ (-1)"+1.

A: The given matrix is a circulant matrix with a special structure. A circulant matrix is a matrix in…

Q: Find gcd(2,3+5i) in Z[i]

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Q: is it T or F

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Q: Let A = {0, {0}, {Ø, {Ø}}}. State whether each of the following statements is true or false: ØCA {0,…

A: Given A=ϕ,ϕ,ϕ,ϕ

Q: Is 1z = 1 for all z ∈ C?

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Q: [1 1 -1] 1 0 Lo Let A 0). Find A-1. (Show all work.) 1

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Q: Let A = [ L 1 -1 4 24 2 37 2 1 J , V = Is v in Nul(A)? Type "yes" or "no". no Is w in Nul(A)? Type…

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Q: The div A at p(1,1,2) for A is .... Å = x²y i + 2 x z j + 4 - z² y k - 3 ( 3 зі зј о

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Q: Let a, b e C. (i) Prove that if |a| <1 and |6| < 1, then a - b < 1. 1- āb

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Q: Let n E Z. Prove that 3 n if and only if 3 | (n² - 1).

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Q: A = 1 - 2 3 k 0 1 2 -k -4

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Q: Let A = {x e Z: 10|x} and let B={xeZ:15|x}. %3D Then * O Bc A O A=B O None of these O ACB

A: Suppose A and B are two sets . Note the following result : If x∈B for all x∈A then A⊆B . If A⊆B and…

Q: Let X={x=(x1, x2, X3, X4) : Xi E {1,2,3,4,5,6,7,8,9,10,11,12} for i=1,2,3,4 such that x1 < x2 < X3 <…

A: Given:X={x=x1, x2, x3, x4 : xi∈1,2,3,4,5,6,7,8,9,10,11,12for i=1,2,3,4 such that…

Q: 1 real number such that a+ E Z. Prove that a" + 1 E Z for any n E N. Let a be any

A: Given that α+1α∈ℤ where α∈ℝ. To prove that αn+1αn∈ℤ for any n∈ℕ , use the method of induction. That…