Question 1 Let A = [2 4 1 7 3 -2 3 0 7-1 -5 4 " write down the cofactor expansion 1 2 -5 of det A with respect to the third column. You do not need to compute. a b C Question 2 Suppose d 9 - - 2a d 2b e 2c-f e f 4. Compute d-5g e-5h f-5i -3g -3h -3i h i Question 3 (True or False) Let A, B be two nxn matrices. If det B = 0, then AB is never invertible. Question 4 Let A, B be two 2 x 2 matrices with det A = 3 and det B = -2. What is det(AB-¹)? Question 5 Suppose A and B are 3 x 3 matrices with |A| = 4 and |B| = 2. Compute (1) |AB|, (2) |A2B-¹|, (3) |3A-¹(B-¹)-¹|. Question 6 Determine if the following subset is a subspace. Justify your an- swer. The set of polynomials p(t) with p(0) = 2 as a subset of P. In other words, H = {p(t) = Pp(0) = 2}. • The set of 2 x 2 matrices with only 0 along the main diagonal as a subset of M2x2. • {(1, 12, 13) Є R³: £1 + 4x2 + 2x3 = 0}. • {(x1, 12, 13) E R³: x1 + 4x2 + 2x3 = 6}. Question 7 Let H be the subset of R4 such that Ha subspace of R4? Justify your answer. x1+x2 x3 + x4 = 0 Is . 21 - x2 x3 x4 = 0 - 3 4 0 7 1 -52 -2 Question 8 Let A = find null space, column space and -1 4 0 3 -1 2 2 row space of A and identify their basis. Question 9 (True or False) If A = b has a solution then be Col(A). Question 10 (True or False) Let A be a 3×4 matrix, then the set of solutions of Ax = b must be a subspace of R4.

Q: The coefficient of k-1 in 1+x (1-2x)5

A: 4) To find the coefficient of xk-1 in the expansion of the x+1(1-2x)5 We use the expansion formula…

Q: 3 5 2 -1 -7 -3 -3 2 1 2

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Q: 1 -1 0 1. Let A=(23¹), B(4). 3 -5 (a) Compute AB and BA. (b) Compute A + B and B + A. (c) If c = 3,…

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Q: 3. Prove that ox + 6x -4X12 is irreducible over Q.

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Q: 6 -1 4 Compute A - 6I, and (6I3 )A, where A = - 5 2 -7 - 4 1 1 A- 613 =

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Q: Use the cofactor expansion to compute the following determinant: 7 7 8 det -1 -10 -10 = 0 8 -2 -10

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Q: 5-A] [CLO 1] Suppose that x, y, z € R. Expand the expression so that it is written without…

A: The expression is given as  (2x+3y+1)(8x-5y+3x+5).

Q: 1 2 Let A = 2 3 1 and C = 0 -2 1 1 | 1 B = -2 Find the trace 3 1 1 1 of (2AB + C)T.

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Q: 22 and 21 21 in the form x + iy. 10 + 3i and z2 = 15 – 5i. Find 22 Let z1 - 21 22 22 21 ||

A: Given complex number is

Q: ) Express the pern 2546 1 product of the generators Wo=(12), `wa=(13), W1# (17) as a (125)(3 4 6 7)…

A: The generator of the permutation group S7 are given by: w2=1,2, w3=1,3, w4=1,4, w5=1,5, w6=1,6,…

Q: 2. Determine tz , t4, and ts for the recursion formula listed below. A) t = 1, t= 3, 4-2 tp-1 -…

A: By using the given formula putting values of n.

Q: If Z1 = 4 + 5i and Z 2 = -1 + I,  express in the form a + bi  (i) Z1 + Z2   (ii) Z1 – Z2

A: Formula used  sum of two complex number is given by  (a1+b1i)+(a2+b2i)=(a1+a2)+i(b1+b2)

Q: Observe the following pattern: 1 + 3 = 22 1 + 3 + 5 = 32 1 + 3 + 5 + 7 = 42  Find the next two lines…

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Q: 1 3 28 (iii) Compute AB (iv) Compute (AB) 3) Let A = and B -1 - 20 4 1 (i) Compute A - 1 - 1 (v)…

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Q: For what value of k, the vectors (-1,0,1), (2,1,A) and (3k+2, 3,10) are lincarly dependent. O A. 1…

A: We have to find for what value of k, the vector -1,0,1, 2,1,4, and 3k+2,3,10 are linearly dependent.

Q: The annihilator operator form for the function e + cos(2x) is:

A: The annihilator operator form for the function ex+cos(2x) .