2. Let A = and B = al. Show that det(A + B) = det A + det B if and only if a + d = 0.

Q: 2 Let A= then det(3A") = 6- -2 - 18 -6

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Q: Let A 1 value of k. [23] A and B = 1 -7 217 k If AB = BA, find the

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Q: Let A=(1, 0), B=(0, 1), C=(1, 1), and D=(r, 1). Verify that R(A, B, C, D) = r.

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Q: Let u = (2,6, 4) and 7 = (-1, -2, - 7). Compute the following: ū. v = vu - ||||²

A: Since you have asked a question with multiple parts in a single request we would be answering only…

Q: 9. Let u = : (4, -1), v = (0, 5), and w = (-3, -3). Find the com- ponents of a. u + w c. 2(u - 5w)…

A: We have u=(4,-1)v=(0,5)w=(-3,-3)

Q: Show that 1 1 1 x2 y? z2 (y² – a²)(z² – a²)(z²? – y²) - - ,4

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Q: Let →u=⟨6,−4,−6⟩ and →v=⟨−5,−3,−6⟩. Compute the following:

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Q: Let ā= (2, 5, 7) and 6 = (1, 3, 4). (a - b) × (a + b) = X (3,16,33) X

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Q: 5. Find all values of h so that is a linear combination of and 4. O h=0 only Oh= 2 only Oh=-2 only…

A: We need to find all values of h so that (h04) is a linear combination of (1-10) and (138) to do that…

Q: 9. Express the gcd(117,213) as a linear combination of s and t such that gcd(117,213) = 117s + 213t…

A: To write the gcd(117, 213) as a linear combination.

Q: Let A= 1 1 Then, ||A||2= -2 Select one: а. 8+45 2 b. 7+/45 C. 10 +45 2 d. 9+45 2.

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Q: verify that (A + B)T = AT + BT

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Q: V ) and A, p 0, and e0. Show that then I',AF0.

A: In the given question  given that γ l=(φ∨ψ) , ∆,φl=θ and and also ψl=θ then we have to prove that…

Q: Let ā = (1, 3,-1) and b (-1,0, 2). What is the y-component of ā x b %3D %3D 1 -1 3.

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Q: Show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

A: To show that the group <a, b | a^5 = b^2 = e, ba = a^2b> is isomorphic to Z2, we need to…

Q: If x=y+qp and f=xy+qp² Express x in terms of q,p and f

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Q: Let (4A)−¹= 2 4 -3 2 Get At.

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Q: Show that is isomorphic to Z2.

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Q: 2. (6 marks) Let a, b, c € Z with a 0. Prove that if a X (a + 2b + 3c), then a Xb or a Xc.

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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: X and  Y are randam variables then XeY is a randam variables.

A: Here, The X and Y are two random variables  Then XeY are also random variables we have to show Here…

Q: Show that (a + b)x = ax + bx for all a, b E F and all x € F¹.

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Q: Let ū = (3, – 4, – 10) and i = (– 7, – 2, 0). – 2,0). - - 2u + 20

A: Given:-u→=<3, -4, -10> andv→=<-7, -2,  0>

Q: Show that A × B ≠ B × A, when A and B are nonempty, unless A = B

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Q: 10.

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Q: Compute u – 3v, u v and u x w.

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Q: 4. Which relation between x and u yields 16+x² = 4/1+u?

A: Given:

Q: -1 1 Let A= 1 Then, ||A||2 = -2 Select one: a. 9+V45 2 b. 8+V45 c. 10+V45 2 d. 7+V45

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Q: 10. Let ā = (1,2) and b= (2,1). Find projz(ā). Then show that ā– projz(ā) is orthogonal to b…

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Q: Let u = , v = and w = . Now, i = is a linear combination of ü, v and w. That is, i = aju + azv +…

A: ANSWER:

Q: Hew.2 Exfress the Summ ef the Vecturs AB and if ACl,-1)_, B (2,022C(-l,3) and D(-2,2).

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Q: Show that if we make the change of variables =x – ct and n =x + ct, then, as in (16.7), a?u ,a²u a…

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Q: Show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

A: We have to show that <a, b | a5 = b2 = e, ba = a2b> is isomorphic to Z2.

Q: Show that (AB)T and BTAT are equal.

A: Now,

Q: Let S={(0,-1,1),(0,-2,-1),(1,1,2)}.

A: S?