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

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

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

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

(a) Construct the consumption matrix Cfor this model.

(b) Compute the matrix (I – C) ¹.

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

   (d) Also compute the equilibrium production levelsfor 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 fromthe mining sector. Compute the new consumption matrix C* and thennew (I – C*) ¹. Interpret the entries of the inversematrix and compare to your answer to part (f) to explain how thechange in the energy sector will affect this economy. .

3. Let L be a line in R² defined by y = mx + b. Thatis, L has y-intercept (0, b) and slope m. In this problem, you willconsider different cases for the line L and and how to reflectpoints in that line. You do not need to multiply out the productsto a single matrix; you can simply leave your answer as a fewmatrices 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) inhomogeneous coordinates will give its image under a reflection inthe 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. Sincewe don’t know what b is (other than b 6 ? 0), the matrix will haveto 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 apoint (x, y) in homogeneous coordinates will give its image under areflection in the line L.

3. Now suppose that L does intersect the x-axis, does so at theorigin, and does so at an angle of ? (measured from the positivedirection).

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

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

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

4. Finally, suppose that L does intersect the x-axis, but not atthe origin, and does so at an angle of ? (measured from thepositive direction). Find a product of 3x3 matrices that whenmultiplied with a point (x, y) in homogeneous coordinates will giveits 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)