6. Consider the subring S = {[0], [2], [4], [6], [8], [10], [12], [14], [16] } C Z18 with respect to addition and multiplication defined in Z18, do the following questions. (a) Does S have the unity? (b) Does S have zero divisors? Which element of S have multiplicative inverses?

Q: Compute the pseudo inverse of A

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Q: . Let a, b € Z so that 4a + 5b is a multiple of 13. Prove that 13 divides 9a + 8b.

A: We have to show that 13 divides 9a+8b To prove this we using 4a+5b divisible by 13

Q: 19 (e) a = 1575, b = -231 (a) 3710

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A: Solution

Q: Let 71 = 3+7i and 72 = 3– 7i and find z172.

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Q: -275 2. Let A 013 -135 Find the adjoint of A.

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Q: a. For real numbers a, b, c, and d, find the product. 01 a 0] b. Based on the form of the product of…

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Q: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

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Q: Let V be all elements of R2 that have the form (a, a2) where a is a real number. Show that (3,9) is…

A: Let V be all elements of R2 that have the form (a, a2) where a is a real number. We have show that…

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Q: Compute the pseudo inverse of A

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Q: Compute: 3] 2 5-3

A: 1.a1a2a3a4+b1b2b3b4=a1+b1a2+b2a3+b3a4+b42. Two matrices can be added iff the order of the marices…

Q: 5. Suppose that GF (2³) is defined by the irreducible polynomial x³ + x + 1. From 4., deduce that…

A: The objective of this question is to find the multiplicative inverse of 101 in the Galois Field…

Q: 3. Prove that if m is a positive integer and x is a real number, then 1 2 m 1 [mx] = [x] + \x + = |…

A: The greatest integer function is one of the most important real valued number theoretic function.…

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Q: 6. Show that there is a one-to-one correspondence between the Cartesian products A x B x C and A x…

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Q: 2 19) Find the value(s) of k such that A = [13] has an inverse. What is the inverse of A?

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Q: 2. Find the LU factorizations of and 20 5 2 -6 3 -13 -3 4 9 16 17 2 4 -6 2 13 9 دن و ان 5 4

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Q: (a) Prove that the set O equipped with the addition and multiplication of complex num- bers…

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