Show that [o 2] A 3 0 has no A = L-U factorization where L is lower triangular and U is upper triangular: Suppose for contradiction that A = and show that there are no choices of a, b, c, x, y, z such that this equation holds.

Q: Use Gauss-Jordan Reduction to find the value of the unknowns in the given set of equations. -7 1 3 +…

A: We can solve this using gauss Jordan reduction method

Q: Find necessary and sufficient conditions on the numbers a, b, c E R under which the polynomial xª +…

A:

Q: In real-number multiplication, if uv = uw and u # 0, we can cancel u and conclude that v = w. Does…

A: To explain: either v=w or give counter example.

Q: Derive the Quadratic Formula, by selecting the appropriate pulldowns for J, K, L, M, N, P, and Q.…

A: Explanation of the answer is as follows

Q: For a E N, a > 1, is a + 1 always invertible, always a zero divisor or sometimes one or the other in…

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Q: For which values of a, b, and m does x + a = b have a solution in Zm?

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Q: Use the given LU factorization to find all solutions to Ax = b 0 5 1 0 0 15 A = 4 1 0 0 -2 |, b 54…

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Q: et A be m x n and let  be in R^n. If A has a row of zeros, show that Ax has a zero entry

A: We have to solve given problem:

Q: The order of the element 3 in Z24 is

A: Given: Z24 Formula Used: The order of an element k in Zn is given by Ok=nhcfk,n

Q: Given constants a,b,c, and d such that  x2+ax+12 has factors of (x+4) and (x+b), and x2+cx-4 has…

A:

Q: Show that A = has two different factorizations of the form A = L · U where L is lower triangular…

A: Given: A=0001. To show: A has two different factorization of the form A=L·U, where L is lower…

Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

Q: Suppose x,y,z are mutually coprime positive integers that solve x2+y2=z2. a) Show why exactly one…

A: Basic properties of modular arithmetic and gcd will be used in solving this problem. The detailed…

Q: The polynomial z + pr + qr + ra + 6 is exactly divisible by each of (z 1). (z - 2) and (z-3). Find…

A: Given,

Q: Assuming x E Z. Prove 2|(x – 6x + 5) → 2}x a|b b = ke Z a 6. = 2 € Z 3 3|6 is True because 4 3+ 4…

A:

Q: Prove that if A ∈ M 2×2(F) is upper triangular, then det(A) equals the product of the diagonal…

A: Let the matrix A be the upper triangular matrix as follows.

Q: 20. Algebra mZOZP = 4r+ 2, MZPZQ = 5r - 12, and MZOZQ = 125. What are m2OZP and MZPZQ? %3D

A: In this question we have to find m∠OZP and m∠PZQ

Q: Find the value of ‘v’ in v2 – 2v -15 = 0, if v is a natural number.

A: Given quadratic equation is,

Q: Find P and D such that P is invertible, D is diagonal and D= P'AP, for 2 -2 3 0 3 -2 0 -1 A=

A:

Q: Use the Factor Theorem to prove that x + c is a factor ofxn + cn if n Ú 1 is an odd integer.

A: To prove the given statement.

Q: If z1 = 3 + 2j , z2 = 4 − 8j, and z3= 2 + 5j find the conjugate of d) z3/z2

A: z1 = 3 + 2j , z2 = 4 − 8j, and z3= 2 + 5j first we find  z3/z2

Q: 18. Let xo < x, < x2, let {yo, y, y2} be given. Find the polynomial q(x) with deg(q) < 2 and such…

A:

Q: 1. Given that r + px+q and 4x +q have a factor (x-r) where p,q and r are non-zero. Show that 4p…

A: topic - quadratic equations

Q: Find d = gcd (270, 105) and hence find integers x , y such that d = 270x +105y .

A: GCD stands for greatest common divisor for two or more numbers.

Q: 7. Show that the equation p(x) = q(x)m(x) + R(x) is equiva- lent to the equation p(x) = m(x) + R(x)…

A: Given equation is  px=qxmx+Rx

Q: Is the polynomial x2 – 4 in the ideal generated by the polynomials x + 22 – 4x – 4, x – 22 – 4x + 4,…

A:

Q: Verify that A + B and AB are upper triangular.

A: Given,    A=32-10-43000    and   B=6-32024003 Recall that, The square matrix is called upper…

Q: Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m,…

A:

Q: 5. Suppose A = PDP¯1, where P is 2 × 2 and D = 9I. Let B = 51 – 3A+ A². Show that B is…

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Find the general solution of у "— 2у "— 16у'+ 32у -0 given that r = 2 is a root of the…

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Q: Suppose x-j, for j-0, 1, 2, 3 and it is known that Find Po123(1.5). P₁1(x)=x+1, P₁2(x)=3x-1, and…

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Q: 2. Show that if x and y are perfect squares (that is, x = m? and y = n2, where m and n are…

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Q: Consider a LU factorization of -2 1 3 A = 1 4 -2 -5 as A=LU. Then L2,1 and U1,1 are respectively…

A:

Q: The Cholesky factorization of 53 is UTU where U is: 3 2 V5 5 -3 V5 1 3. 0 1/5] V5 5 3 0. 1

A: The given matrix is,5332

Q: Prove that {4a + 126 : a, b E Z} = {4c: c € Z}.

A:

Q: 5. Let a, b, N E Z where N > 0. Prove that [ab] = [[a][b], where [r] denotes the remainder of r…

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Q: 2. Prove that if alc and a + b = c, then alb. 3. Show that if 2a + 1 is a multiple of 5, then 14a² +…

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Q: Prove that the area of a square whose diagonal has length d is A = A = 1/2d². A square with a…

A: A square has diagonal length of d, we have to find its area.

Q: 8 1 4 9 1 1 1 4 7 10 3 7 7 11 1 What is the mode? If there is more than one mode, enter your answers…

A: Mode: mode may be defined as the maximum frequency of the observation is called mode.

Q: Find the canonic analytical expression of the following quadratic form q: R³ → R, q (x) = 6x,²+x,² –…

A: Given quadratic equation is…

Q: Exercise 13.4.7. Fill in the blanks to prove that given any integers p,q with 0 . • By multiplying…

A: The proof is by contradiction: so given integers p, q with 0 &lt; p, q &lt; n, we suppose sorp = rq…

Q: Q1) Compute the products using the definition as a linear combination. 8 3 -4 1 2 1 5 1

A: As per the definition of linear combination abcdef.x1x2x3=x1.ad+x2.be+x3.cf

Q: Find the domain of gof, iff(x) =x² and g(x) = Vx-6. %3D O a.l-, - võJU(võ. x) O b. (-30, -6]U[6, ¤)…

A:

Q: use the convolution theorem to find L'{2+1 -1 4 Specifically s²(s²+1)

A: Given,        L-14s2s2+1=

Q: Given the set of multiples of 5 defined by 5Z:= {m: m = 5n for all n e Z} together with the…

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Q: 10. Suppose that A in has the SVD A= USVT Use this factorization to solve Ar =

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Show that Z[√−D] is not a unique factorization domain, where D is squarefree and positive

A: Given: To be prove Z-D is not unique factorization domain.

Q: Use the Factor Theorem to prove that x + c is a factor of xn + cn if n ≥ 1 is an odd integer.

A: Let f(x)=xn+cnNow take,x+c=0i.e.x=-cNow, considerf(-c)=(-c)n+cn          =0          (Since, n is an…

Q: For all a and b in Z, a - b is a factor of an - bn. (Hint: ak+l - bk+I = ak(a - b ) + (ak - bk)b.)

A: we have to prove that For all a and b in Z, a - b is a factor of an-bn