You are building a company centered around a cool design for a T-shirt. Step 1 The Design - production costs You can produce 25 T-shirt for $12 each and 275 T-shirts for $7 each. Figure out the fixed costs and write the equation linking the production cost of the T-shirt Pt to the number of T- shirts Nt. Step 2 Selling price You want to sell the shirts for P-$20 each. Write the equation for the selling revenue from the T-shirts, St. Step 3 Diversification Obviously, you cannot have a profitable company selling just T-shirts. So you decide to put your design on other products to increase the rentability of the whole project. In addition to T-shirts, you will sell mugs and keychains. You can sell mugs for $13 and they cost $10 to make. Keychains are cheap to make, they cost $2 per keychain plus $2 of fixed cost. Keychains can be sold for $5. Write the equations for the production costs and selling revenues of the mugs Pm, Sm and the keychains Pk, Sk. Step 4 How much can you sell? Realistically, it will be much easier to sell keychains on campus, so you expect to sell 10 times the amount of T-shirts. You figured out researchers drink a lot of coffee so you show up at every conference on campus and sell a large quantity of mugs: half the amount of keychains plus twice the amount of T-shirts. Write the equations for the number of mugs Nm and keychains Nk. Step 5 PROFIT How many shirts, mugs, and keychains do you need to sell to break even? Solve the system using the method of your choice. Finally, you are doing all this to get a profit, right? How many T-shirts, mugs and keychains, do you need to sell for your total benefit to be $5,000? Solve the system for Pt Pm Pk Nt Nm Nk using Gauss elimination. Write the same system for Nt, Nm and Nk unknowns only, solve that system with Gauss elimination and Cramer's method. 2. Inverse of a matrix We will be starting from the 3x3 matrix A determined in part 1 of the homework challenge as the system to solve for Nt, Nm, and Nk. Find A-1, the inverse of A, using Gauss-Jordan elimination method. You can check your result with the inv function in Python. 3. Multiplication of matrices Using A-1, find the result of the system from part 1 by matrix multiplication. Compare with what you got in part 1 using Gauss elimination. 4. Eigenvalues and eigenvectors Given E the following matrix: Find the eigenvalues of E (without using the eig function), that 85 -28 -28 E -10 -11 -11 -46 -2 -2 is find the characteristic polynomial and solve the characteristic equation. Find the eigenvectors of E (without using the eig function). More precisely, what is expected is: - From the eigenvalues found in 4, build the system of equations to solve. - Use Gauss elimination to get the three components of the eigenvector. - Perform the same thing for each of the 3 eigenvectors, changing the eigenvalue. - Normalize the eigenvectors. . Given the following matrix K, find the eigenvalues and eigenvectors of K. K= 5 -10 -5 2 14 2 -8 6 T