Let R= [0, a, b, ct + a 6 ( a 6 2 a O 2 G G b C 0 a & Sola ه حب a 0 a b ( a b ( States all Subrings of R States all ideal of R ? b O ( a b (

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

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Q: N 3 -1l Find the adjoint of A: A =| 1 6 3.Then verify that A (adj A)= (det A)I. 2 -4 0

A: Verification:Compute adj(A) as follows.

Q: -5 7 Eхample 61. Find the adjoint of A = -2 and show that A (adj. A)=|A| I.

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Q: Let 2 = {a, b, c}. Write down all possible o-algebras on N.

A: Given: Ω=a,b,c

Q: Find JA|, |B|, A + B, and |A + BỊ. Then verify that |A| + |B| # |A 1 6 [o 1 -1 A = 1 -1 0 B = 6 1 1.…

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Q: If (R,+,.) be aring with identity and A is : ideal of R, then the identity element of R/A is * O A+1…

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Q: O Let (eto-) is aring of complex-numberrs and M be aringof2x2 matrices aud desined as sollows. g e…

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Q: 2) Let R' real numbers, define = x,Y1 + 3x2y; + 5x3y3 for x = (x, X2, X3) and y (y,Yz, Y3). Is…

A: Given x=x1,x2,x3, y=y1,y2,y3 in ℝ3. Define x,y=x1y1+3x2y2+5x3y3

Q: Z3-22 If z1 = -4 - j5, Z2 = -3 + j2 , and z3 = 2 - j3, what is Re Note: z* is the complex conjugate…

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Q: Let a, b, q, r E Z and assume that a and b are not both zero and b and r are not both zero. Show…

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Q: If 7,=-4-J5 Z2= -3 +j2, and Zg =2-3, what is Re (23-2 7, + 23 Note : z* is the complex conjugate Of…

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Q: |1 0 -3 8 7 45 0 1 a Let M(a, b) = where a and b are real numbers. a la 0 b 1| a) Evaluate M(a, b)…

A: Ma,b=10ab745-38b01aa0b1 Expanding the determinant along column 2, we get  Ma,b=10ab745-38b01aa0b1…

Q: + Let A = B = 1 4 Then find: AB & BA - 2

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Q: Ex: tet R = {o, a,b,c} a G O a b ( وه هه 2 2 G 6 ( 0 a C b a O a b a b 2/2 ( в с @ Show that (B,…

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Q: Use the adjoint of A = [1 [1 -5 -2 27to 1 to find A-¹ 0 2

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Q: nd a, b e R such that 4 5 a + bi 1- 2i i - 2 i- 2

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Q: Let and 001 1 0 0 2 1 A = 0 If the entries of A and v belong to Z3, what is the third entry of Av? 2…

A: Recall matrix multiplication and modulo arithmetic.

Q: 12. Which of the following polynomials is reducible over Q а) 4x3 + x — 2 b) Зх3 — 6х? +х - 2 c) 5x3…

A: Reducible, Irresistible

Q: Take a look at the operator defined as follows: ★: 2 × 2 → Z x+y=x+3y 1) Does the operator have a…

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Q: 2. Z [x] is a principal ideal domain. a) True b) False O a) True O b) False

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Q: 112х+ 7 G Зx+ 16 E

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Q: Which ones are echelon form, reduced echelon form or not echelon form?

A: To find which of the matrices are in echelon form, reduced echelon form or not in echelon form.

Q: Solve the follawing yetem using augmented ito Matrisc M! C + + 32=1 *+34-2= 3 + Z=7 2x+

A: The given system of equations is x+y+3z=12x+3y-z=35x+7y+z=7 Solving the given system by converting…

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (x, y)…

A: Let u=(x,y) and v=w,z be any two elements of W, where w,x,y,z>0. Use the definition of addition…

Q: prove that two vectur ae orthognal A = x + 4y+32 B =4x+2y-42

A: See the attachment

Q: 9. Let D be an integral domain such that (x + 1)² = x² + 1 in D[x]. Char D = а) 0 b) 1 c) 2 d) 3 O…

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Q: 4. Find the value(s) of a for which and are orthogonal: a) (2,-4,5) and (a +1,3,-a +3) b) -…

A: As per guidelines we must answer only one of multiple questions given. Please resubmit the remaining…

Q: For any vectors u, v in R', we havelu + ū|? = [u|2 + [ū]2 True False

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Q: 6. show that the set A = {P= (₁4₁2) (R³ | 1²4 + √²2+x2 is an open out.

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Q: Find JA|, IB|, A + B, and |A + BỊ. Then verify that |A| + |B| # |A + B|. 15 0 1 -1 A = 1 -1 0 B = 5…

A: topic - determinant and matrices

Q: Rinding Construcding Al or 2 3 A z → (AII) (AIT)

A: The given matrix is: A=12310-1231 To find its inverse.

Q: Show that I+J is an ideal of L where I+J ={x+y:x∈I, y∈J}.

A: Given that and  are two ideals of a ring Ideal Test:A nonempty subset A of a ring is an ideal of…

Q: 4r and use the result to Identities Prove that V 12 prove that V. V |r|6*

A: To prove:∇1r4=-4rr6 also using this identity prove that ∇·∇1r4=12r6.