Thora, David, and Binky are three small businesses in Elwood City. Thora makes pastries, David makes ice cream,
and Binky grows strawberries. They each share ingredients as follows.
• Thora keeps 20% of her pastries, and sends 40% and 40% to David and Binky, respectively.
• David keeps 40% of his ice cream, and sends 30% and 30% to Thora and Binky, respectively.
• Binky keeps 20% of his strawberries, and sends 50% and 30% to Thora and David, respectively.
The above information is encoded in the following 3 x 3 matrix.
$$A = \begin{bmatrix} 0.2 & 0.3 & 0.5 \\ 0.4 & 0.4 & 0.3 \\ 0.4 & 0.3 & 0.2 \end{bmatrix}.$$
Use matrix to complete the below tasks.
(a) Write a few sentences interpreting what the rows and columns of A represent.
(b) Solve the equation AX = X for X, where X is a 3 x 1 matrix. Write a few sentences interpreting what the
solution X represents.
(c) Use determinants to show that the matrix 2I3 - A is invertible.
(d) Consider the equation AX = 2X, where X is a 3 x 1 matrix. Use the result of (b) to show that the only solution
to this equation is X =
$$\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$. Write a few sentences interpreting what this conclusion represents.
To earn full credit, you must solve the equation in part (d) using matrix multiplication and inverses to isolate X.
Solutions involving only linear systems will not earn full credit.
