The zero-product principle states that if AB = 0, then________.

Q: a) Determine all value(s) of x1, ×2, X3, and x4, such that 3 0 5 x1 x2 x1 x2 3 -4 х3 х4 x3 x4 5 -4

A: first, the matrix can be multiplied when it is in the form a x b and b x c…

Q: determine whether multiplication by A is one-to- one by computing the nullity of A and then applying…

A: Let (x, y) be in nullspace of A.  Then, A(x, y)t = 1-22-4-36xy = 000     => x - 2y = 0…

Q: z -7 2. prove that: Im(z)= 2i

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Q: 1. f¹ B-A. 2. f is bijective. Suppose f: A→ B. Prove that the following statements are equivalent.

A: A bijection, also known as a bijective function, one-to-one correspondence, or invertible function,…

Q: x and y. Prove that if alb and alc, then a|(xb+ yc) for any integers

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Q: If det(A)=0, then A has a row consisting entirely of zeros. Select one: O True O False

A: Since an entire row or column of a matrix A has all zeros, then each permutation in the calculation…

Q: Let G = <a, b | a2 = b2 = (ab)2>a. Express b2abab3 in the form biaj.b. Express b3abab3a in the…

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Q: A = 6. -3 -3 -4 3.

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Q: Prove that 1+1=2

A: To prove: 1 + 1 = 2

Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: - BJ -1]. prove prove A²-A-51 = 0 3 If A =

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Q: Find all (real) values of k for which A is diagonalizable

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Q: Prove that no element of ℤ/n is both a zero divisor and a unit.

A: To Determine :-  Prove that no element of ℤn is both a zero divisor and a unit.

Q: a Prove that I = { : a,b,c,d are even integers} is an ideal of M2(Z). d

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Q: A−1 in the block form.

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Q: Prove that (A-1)-1 = A. %3|

A: Let A be inverse of matrix A. Then, by definition We know that

Q: Prove that for every real numbers a, b, and c, where b a is not equal to zero, if a +b = c, then=+1

A: Given,            For every real numbers a,b and c, where b is not equal to zero.            if…

Q: If A has a row or column consisting entirely of zeros, then Select one: a. A is nonsingular b. A is…

A: Given: A is row or column consisting entirely of zeros .

Q: (a) Given A = use the Cayley-Hamilton theorem to find (1) A¹-3A6+ A¹+3A³-2A²+31 (ii) A¹, where k> 0…

A: (a) Given A=2011 To use the Cayley-Hamilton theorem to find (i). A7-3A6+A4+3A3-2A2+3I (ii) Ak, where…

Q: Prove that there exists two integers a and b such that a+b > ab

A: We want to prove that there exists two integers a and b such that a+b>ab

Q: Prove: | a + b | <= | a |  + | b | for all a's in the reals.

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Q: .ABCDE is apentagon.Prove that the resultant of →AB,−→AE,−−→BC,−−→DC,−→ED and AC is 3AC

A: Here we have given a pentagon ABCDE. Consider the Figure for the solution. We have to prove that…

Q: Find the minimal polynomial of the following algebraic numbers. 3 22 (1 + i)/2; - (1+ i)/v2;

A: Since you have asked multiple subpart question, we will solve the first three subparts for you. If…

Q: Find all (real) values of k for which A is diagonalizable

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Q: Find all positive integers a, b such that a | b^2-b+1 and b | a^2-a+1

A: Solution :-

Q: 9- Z is Principal Ideal Domain. a) True b) False O a) True O b) False

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Q: For an integer t, prove that gcd(2t + 1, 9t + 4) = 1.

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Q: Let z= a + ib is it true or false that |z²| = (Re(z)) + (Im(z)) where Re(z) is the real part of z…

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

Q: There exist infinitely many polynomials of degree 3 which satisfy p(0) = p'(0) = p"(0) = p"(0) = 1.…

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Q: Let jEZ (a) Prove directly the statement : If 3j, then 3j²

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: if z1-z2 and z1/z2 are both real then prove that two must either real or equal to each other

A: Suppose z1-z2 and z1z2 are both real. We need to prove either z1 and z2 are real or equal to each…

Q: There exists two odd integers whose average is odd. 1 1 1 There exists distinct positive integers a,…

A: (a) Consider two odd integers 2m+1, 2n+1 such that m+n is even. Now, Consider the average of the…

Q: Prove that every perfect square is either a multiple of 5 or of the form 5j - 1 or 5j + 1.

A: Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap −…

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: Given de - and d, - Then do and d, are not G- conjugate if Gis Non of these -0.25 105. 0.25 -0.5 -0…

A: We want to find the matrix G for which given vector are not G conjugate

Q: If a, b, c are integer powers of 2, prove that equation a° + 64 .3 + 64 = c5

A: Given: Let's take a,b,c are integer powers of 2. Prove that a3+b4 =c5  does not have a solution.

Q: Let co, C1,..., C5 be real numbers satisfying C1 Co 2 C2 C3 C4 + 4 C5 C7 C6 긍 = 0. 7 - 5 6. Prove…

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Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…

Q: 1) a) Are G1 and G2 Isomorphic? Why? b) Are A and B Isomorphic? Why? A

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Q: Га Let a be a non-zero real number. Using Gauss-Jordan Elimination, the inverse of 0 Lo 1 01 0 is 1…

A: Given a00010011  We need to find inverse of above matrix by Gauss-Jordan Elimination method.

Q: a Let A Show that A is nonsingular if and d only if ad – bc + 0.

A: Known fact: A matrix is said to be invertible (nonsingular) if its determinant is nonzero.

Q: let a>0. prove that if x is a number such that |x-a| a/2

A: Given a>0 and x is a number such that x-a<a2 We will use the property If y<b , then…

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

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Q: Prove that given a and b, with b> 0, ther exist unique integers q and r satisfying a- qb•r in the…

A: We will prove the given statement.

Q: Find bases for the null space and row space of A. 3 4 6. -10 -8 -4 -16 A = -1 -1 0 -2 3 4 7 13 14

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Q: 0. Prove that (i) |4' ||, =||4||,, and (ii) ||4' A|| -||4|-

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Q: If z = what is the conjugate of 2/프+4/. 4 z? O 0.913 – 0.426i O -0.913 + 0.426i O -0.913 – 0.426i O…

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Q: If a, b, and c are distinct real numbers, then the polynomials (r b)(x- c), (r- a)(x - c), and (r -…

A: In this question, we have to check the set of polynomials are linearly independent or not.

Q: Let A = {n ℤ | n = 5r for some integer r}and B = {m ℤ | m = 20s for some integer s}.Determine which…

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Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.