Question

1. Suppose that A is a 6 x 6 matrix that can be written as a product of matrices A = BC where B is 6 x 4 and C is 4 x 6. Prove that A is not invertible.

2. An economist builds a Leontief input-output model for the interaction between the mining and energy sectors of a local economy using the following assumptions:

• In order to produce 1 million dollars of output, the mining sector requires 0.1 million dollars of input from the mining sector and 0.5 million dollars of input from the energy sector.

• In order to produce 1 million dollars of output, the energy sector requires 0.6 million dollars of input from the mining sector and 0.2 million dollars of input from the energy sector.

(a) Construct the consumption matrix C for this model.

(b) Compute the matrix (I – C) ¹.

(c) Find the equilibrium production level when the final demand is d = (10, 40).

   (d) Also compute the equilibrium production levels for final demands (1, 0) and (11, 40).

(f) In light of your answers to parts (c), (d), and (e) above, interpret the entries in

the matrix (I – C) ¹.

(g) Suppose that due to the growth of green energy companies, the energy sector requires only 0.3 million dollars of input from the mining sector. Compute the new consumption matrix C* and then new (I – C*) ¹. Interpret the entries of the inverse matrix and compare to your answer to part (f) to explain how the change in the energy sector will affect this economy. .

3. Let L be a line in R² defined by y = mx + b. That is, L has y-intercept (0, b) and slope m. In this problem, you will consider different cases for the line L and and how to reflect points in that line. You do not need to multiply out the products to a single matrix; you can simply leave your answer as a few matrices multiplied together

1. Suppose that L is the x-axis.

1. What is m? What is b?

2. Find a 3x3 matrix that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

2. Suppose that L does not intersect the x-axis.

1. What is m?

2. Find a 3 x 3 matrix that will translate L to the x-axis. Since we don’t know what b is (other than b 6 ≠ 0), the matrix will have to include the unknown b.

3. Find another 3 x 3 matrix that will translate the x-axis to L. Again, this matrix will have to include b.

4. Find a product of 3 x 3 matrices that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

3. Now suppose that L does intersect the x-axis, does so at the origin, and does so at an angle of θ (measured from the positive direction).

1. What is b? By trigonometry, m = tan(θ).
2. Find a 3 x 3 matrix that will rotate the line L to the x-axis.

3. Find another 3 x 3 matrix that will rotate the x-axis to the line L.

4. Find a product of 3 x 3 matrices that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

4. Finally, suppose that L does intersect the x-axis, but not at the origin, and does so at an angle of θ (measured from the positive direction). Find a product of 3x3 matrices that when multiplied with a point (x, y) in homogeneous coordinates will give its image under a reflection in the line L.

4. Let A be an n x n invertible matrix. Prove that

det(A^-1 ) = 1/ det(A)

Answer 1