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QUESTIONS Problem 3. Let P, Q be nxn matrices with PQ = QP. Suppose that is nonsinsingular and veR" is a nonzero eigenvector of P. Determine which of the following statements is True. e: v and Qü are eigenvectors of P with the same eigenvalues. 12: v and Qü are eigenvectors of P with distinct eigenvalues. T: Qü is not an eigenvector of P. V: None of the other answers. e оооо
The inverse demand curve for product x is given by px=20−4·qx+2·py where px represents the price in dollars per unit, qx represents the rate of sales in pounds per week, and py represents the selling price of another product y in dollars per unit. The inverse supply curve of product x is given by px=10+2·qx Determine the equilibrium price and sales of X Let py=$10. Determine whether x and y are substitutes or complements
#9. If P and Q are two transition probability matrices with the same number of row (and hence columns) will PQ necessarily be a transition probability matrix? Justify your answer.
Let P and Q be two projectors, such that PQ zQP prove that (PQ) is a projector and (a) CLPQ) = c(p). ndo) (6) Null (PQ) = Null P + Null Q
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
step by step please.
30. Let p and q denote quaternions and let a,b E R. Show that (b) (ap + bq)apbq (c) N(q) = qq* = qq (d) pq)* = q*p* [Hint: First show that (iq)* =-qi, (jq)* =- (kq)* =- (b).] (e) N(pq) -Np)N() [Hint: (c) and (d).] of k. q J, and g K, and then use
1. Let P be any point on the line: 1:= (4,8, -1)+(2,0,–4),TER. Let Q be any point on the line: 12: .X-7_1-2_2+1 -6 Find the Cartesian equation of the plane formed by all the possible midpoints of the line segment PQ.
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [
3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ]
(i) Calculate A + B,
(ii) Calculate AB
(iii) Calculate the inverse of B,
(iv) Calculate the determinant of C.
(b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2)
and (5, −1, 4) respectively.
(i) Show that P Q~ =...
5.104. Let p and q be irreducible elements of a PID R. Prove that R/(pq) = R/(p) x R/(q) if and only if p and q are not associates.
Carefully draw the line segment PQ that connects P=(4, 5, -3) and Q=(0, -4, 2) . Include dotted vertical lines from the xy-plane to P and Q to show perspective. Find the distance between P and Q, from the previous problem. Then find the coordinates of the midpoint of the line segment PQ . Let u= -3i+5j+7k and v= 10i+j-2k . Show that u × v is orthogonal to the vector v .