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: n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0

A:

Q: Icm(a, b). Prove that Let a and b be positive integers, and let d = Za × Z, = Za × Zm· gcd(a, b) and…

A:

Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

A:

Q: 2. Let A M, (R) be given. (a) If AA¹ On, prove that A-0₁. Does this hold if A € M₁(C)? (b) Suppose…

A: The objective is to prove the two given parts. Symmetric matrix: A square matrix is said to be…

Q: Let A = {n ∈ Z; n is odd} and B = {n ∈ Z; 3n 2 + 7 is even}. Prove that A = B.

A: We have to prove A is contain in B and vise versa

Q: Let 0 < a < 1 be a real number. Define a₁ = 1+a, Prove that Vn € N, 1 ≤n, an+1 = 1<an< 1 - an + a, -…

A:

Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

A:

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: (b) Let A¡ = [-i, 1] for every positive integer i. Find U A¡ and N1 Aj.

A: Given that Ai=-i,1 for every positive integer i. That is, A1=-1,1, A2=-2,1, A3=-3,1 and so on. To…

Q: Let  A = (1 1 1, 1 2 2) Find R(A), R(AT), and N(A)

A: Given,A=111122the reduced row echelon form of AMake zeros in column 1 except the entry at row 1,…

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: Prove that (a; + b;) = a; + i= 1 Σύ. i=1 i=1

A:

Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: 2 Q) Let A € M₁ C4) such that A² = A ENA show that det (I+A) = 2

A: If minimal polynomial is totally factorizable that is product of linear factors. Then matrix is…

Q: Assume that 0  ≤ a1 ≤ 1 and that  an+1=3√(1+an)/2 for all n ≥ 1. Prove that (a)0  ≤ an ≤ 1 for all…

A:

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

A:

Q: 28. Prove by induction that i(i – 1) (n+1)n(n– 1) i=1 holds Vn E N,n 2 1.

A:

Q: Prove that: n(n+ 1) r = 2 i=1

A: ∑i=1nr=1+2+3+....................+nhere it is a arithmatic progression sum of n terms of arithmatic…

Q: Prove that P(Ø) = 0. Hint: Let A, = Ø for all i in equation (1.3.3).

A: Consider that

Q: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

A:

Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

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

A: A matrix is singular if its determinant value is zero. So we find the determinant value of the given…

Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b € N, and odd c€ N such…

A: Since n is even and non-zero . Since n is non zero ⇒ n is either positive or negative. Let's write…

Q: (f) Prove by induction that for any positive integer n, n(n +1)(2n + 1) j=1

A:

Q: Let S = {n + m√2 : n, m ∈ Z}  To prove that the sum of any two elements of S is an element of S.…

A: Given: S=n+m√2:n,m∈Z  (a) Let a+b2∈S for all a,b∈Z and let c+d2∈S for all c,d∈Z Add both these…

Q: Prove Theorem . k(AB) = {kA)B = A{kB)

A: Proof: Given:   Let A and B are matrices of the same order and let k be the scalar.  To prove:…

Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

A:

Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

A:

Q: 1. Let z E C. Prove that Re(z?) = [Re(z)]? - [Im(2)]² and Im(z²) = 2 Re(2) Im(2)

A:

Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1…

A: The Fibonacci sequence can be defined as a series where a term of the sequence is the sum of the…

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: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b e N, and odd c e N…

A: Consider the given condition,  n∈ℤis a non-zero and even. The objective is to prove that there exist…

Q: a) Prove that ε n(n+ 1)(2n + 1) Σ 6 i=0

A:

Q: Let n € Z. Prove that if 3 /n, then 3 | (2n² + 1).

A:

Q: Given a positive integer n, let a2,n a2,n–1 a3,n–2 a3,n–1 a3,n An = An-1,n аn-1,2 ап-1,n-2 аn-1,л-1…

A: For given a positive integer n, let An=00⋯00a1,n00⋯0a2, n-1a2,n00⋯a3,n-2a3,n-1a3, n⋮⋮⋱⋮⋮⋮0an-1,…

Q: Determine all values of s so that 1 A = | 1 0 1 1 S is nonsingular.

A: Given matrix: A=s101s101s

Q: Let a; b; c 2 N such that a + b = c. Prove that gcd(a; b) = gcd(a; c) = gcd(b; c).

A: The given problem is to prove the following condition for given equation a+b=c, where a,b,c belongs…

Q: Fix n e N. Prove that gcd(5n + 3, 3n + 1) = 1

A:

Q: 3. Prove that A=I-N+N²_N³ for A & N defined as follows: N=A–I (identity), where 1 0 0 & I=|0 1 0 1…

A:

Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n > 3, 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 an express an…

A:

Q: Let -1 0] A= 0 2 2 -3 0| Prove that A'A=I.

A:

Q: 4.3 (i) Let X ~ N (µ, o), and for 0 x1-a) = a, so that P(x, <X< x1-a) = 2a. Show that x, = u +o$²…

A: here X is normal distribution. 0&lt;alpha&lt;1

Q: 6. Let K be compact in R and let c E R. Prove that there exists a point a € K such that |c – a| =…

A: Given- Let K be compact in R and let c ∈ R.  To Prove- Prove that there exists a point a ∈ K such…

Q: Find the value of k such that A is singular. 1 0 5 5 -1 0 2 5 k k II

A: Given query is to find the value of k for which matrix A is singular.

Q: Exercise 3.14. (i) Prove that, for n > 1, it follows that =) 1, it follows that 1 1+ 13 1+ 23 1 n3…

A: Prove that : (i) for n≥1, it follows that         1+121+122 . . . 1+12n&lt;52 (ii) for n≥1, it…

Q: 1. (a) Let A = {x ER: x > 0 and x² < a}. If sup A = B, prove that B2 = a. %3D (b) N is the set of…

A:

Q: Show that Vn 2 0 E-o(2i + 3)3 = (n + 1)3"* %3D

A:

Q: Let x and y be coprime positive integers. Prove gcd(x + y, x² + y²) E {1, 2}.

A:

Q: 1. Suppose a e Z is an integer. Prove that a is odd if and only if a³ is odd.

A: according to our guidelines we can answer only three subparts, or first question and rest can be…