Linear Algebra Answers

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)

Determmine value of k such that

Kx+y+z=1

X+ky+z=1

X+y+kz=1 has a) no solution b) unique solution c) more than one solution

2a. Solve the quadratic equation

3z² + (a - i)z + 3i = 0.

b. Solve the following system of equations of complex numbers:

z + iw = 1 + 2i

z - w = 1 -2i

For which value of k will the vector v1=(1, −2, k) in R

3

be a linear combination of

v2 = (3, 0, −2) and v2= (2, −1, 5)?

Show that T(x1, x2, x3, x4) = 3x1_7x2+5x4 is a linear transformation by finding the matrix for the transformation. Then find the basis for the null space of the transformation.

Let V be a vector space of 2×2 matrices over R. Show that the set S defined by S={(a,b)(c,d)belongs to V :a+b=0} is a subspace of R

Let ( u1,u2,...un) be an orthogonal basis for a subspace W of R^n and let T:R^n-->R^n be defined by T(x)=proj W(x). Show that T is linear transformation.

Show that T(x1,x2, x3,x4)= 3x1 -7x2+5x4 is liner transformation by finding the matrix for transformation. Then find a basis for the null space of the transformation

Let T:R^n--> R^m be a linear transformation and let ( v1,v2,....v3) be a linearly dependent set. Show that the set ( T (v1),T(v2),....T(vn))is also necessarily linearly dependent.

find the sum and product of eigenvalues of the matrix

[1 2 3

-1 2 1

1 1 1 ]