Linear Algebra Answers

If we have a linear equation what is the power of the variable(s) involved

True or false :3Z= Z+Z+Z when Z is the matrix

Let 0 Zero be the zero matrix. Write down the matrkx if of size 2×2

Show that the set {1 − 𝑥, 3 − 𝑥 2 , 𝑥} spans 𝑃2

Determine the basis for the set of vectors lying on the plane 𝑊 = {(𝑥, 𝑦, 𝑧): 3𝑥 + 𝑦 − 5𝑧 = 0}

Find the nullity and the range of the space spanned by 𝑢 = (1,0,2), 𝑣 = (−3,1,1), 𝑤 = (0,1,4) and 𝑥 = (−1, −3,5)

Find the dimension and a basis for 𝑊 = {(𝑥, 𝑦, 𝑧, 𝑤,𝑡): 𝑥 + 𝑦 + 𝑧 + 𝑤 + 𝑡 = 0, 𝑥 − 𝑦 + 𝑧 − 𝑤 + 𝑡 = 0}

Give an example of a 1×2 matrix

1. Agriculture A fruit grower raises two crops. apples and peaches. Each of these crops is sent to three different outlets for sale. These outlets are The Farmer's Market. The Fruit Stand, and The Fruit Farm. The numbers of bushels of apples sent to the three outlets are 125,100. and 75. respectively. The numbers of bushels of peaches sent to the three outlets are 100. 175. and 125. respectively. The profit per bushel for apples is$3.50 and the profit per bushel for peaches is $6.00. (a) Write a matrix A that represents the number of bushels of each crop i that are shipped to each outlet j. State what each entry aij of the matrix represents. (b) Write a matrix B that represents the profit per bushel of each fruit. State what each entry bij  of the matrix represents. (c) Find the product BA and state what each entry of the matrix represents.

A company produces three products which are interdependent. These are A, B and C. The flow of inputs and outputs between the products is represented in the table below:

Inputs (in thousands of units)

A B C Final demand

Outputs

(in thousands of units) A

B

C 40

60

80 65

130

65 75

75

25 20

60

80

Required:

i) Derive the technical coefficients matrix (3 marks)

ii) Determine the Leontief’s inverse matrix (12 marks)

iii) Compute the output level for each product if the final demand for product A increased by 2000 units, that of product C decreased by 1,000 units and the final demand for product B remained unchanged. (5 marks)