Can someone help me with this please?

Matrix assignment

1.) The Ace Novelty Company received an order from Magic World for 900 Giant Pandas,1200 Saint Bernards, and 2000 Big Birds. Aces management decided that 500 Giant Pandas, 800 Saint Bernards, and 1300 Big Birds could be manufactured in the Vancouver plant and the remainder of the order could be filled by the Toronto plant.Each Panda requires 1.5 square yards of plush, 30 cubic feet of stuffing and 5 pieces of trim. Each Saint Bernard requires 2 square yards of plush, 35 cubic feet of stuffing and 8 pieces of trim. Each Bird requires 2.5 square yards of plush, 25 cubic feet of stuffing,and 15 pieces of trim. Finally, the plush costs $4.50 per square yard, the stuffing costs$0.10 per cubic foot and the trim costs $0.25 per unit.

a) Represent by a production matrix the production of stuffed animals by location(locationstuffed animal).

b) Represent by a matrix the component usage in the production of the stuffed animals(stuffed animalinputs).

c) Find how much of each material must be purchased in each plant.d) What is the total cost of materials occurred by Ace Novelty in filling the order?

2.Determine whether the system has infinitely many solutions or only trivial solutions

a) 3w +5x - 4x + 2z = 0 b) x+ y+ z = 0 c) 2x + 5y =0

7w - 2x + 9w + 3z = 0 x - y = 0 x + 4y = 0

x -2y - 5z = 0 3x - 2y = 0

3.) A manufacturer produces three products, A, B and C. The profit for each unit sold of A,B and C, are $1, $2 and $3 respectively. Fixed costs are $17,000 per year and the cost of producing each unit of A, B and C are $4, $5 and $7, respectively.Next year, a total of 11,000 units of all three products is to be produced and sold, and a total profit of $25,000 is to be realized. If total cost is to be $80,000, how many units of each of the products should be produced next year?

4)The holdings of stock of individuals I and II in companies A, B, C, D are given by the matrix

A 200 1000 500 1000

B 1000 500 2000 0

Denote the stock price of stock A by p_A (and similarly for B, C, and D). If the vector ofnstock prices (per share) is (p_A, p_B, p_C, p_D)=(24, 47, 150, 14), calculate the value of thestock holdings of both individuals.

5)If A is a (nn) matrix and x is a (n1) vector. Show that xAx = xBx, where B = (A +A)/2

6)Determine whether z = 4x² + 4xy + 3y² subject to x 2y = 0 is either positive or negative definite.

7)Express each quadratic form below as a matrix product involving a symmetric coefficient matrix

q = - u² + 4uv - 6uw - 4v² - 7w²

q = 3u²- 4uv + 7v²

*From the above quadratic forms, determine if they are positive or negative definite

8)Consider the following system of equations

2x₁ + x₃ = 4

5x_{1 +} 3x₂+ 4x₂ =2

8x₁ + 6x₂ + 7x₃ = -2

a) Write down the expansion of the determinant of the coefficient matrix in terms of the second column. Be specific about the cofactors.

(b) Does the system have a unique solution?

(c) What is the rank of the coefficient matrix? Use the result from (b).

(d) Write down the augmented matrix, find its rank and comment on the existence of solutions.

e)Find the rank of the coefficient matrix (use either the theorem about the largest minor or the elementary row operations).