Use matrices, including matrix products and powers of matrices, to model and solve problems, e.g. costing or pricing problems, squaring a matrix to determine the number of ways pairs of people in a communication network can communicate with each other via a third person.

A communication network is represented by adjacency matrix $M$, where entry $M_{ij}$ equals 1 if person $i$ can send a message directly to person $j$, and 0 otherwise. To find the number of ways two people can communicate via exactly one intermediary (a third person), we must examine the matrix $M^2$. Which option correctly describes what the $(i,j)$ entry of $M^2$ represents in this context?

A small bakery produces three types of bread: sourdough, rye, and wholemeal. The table shows the quantities (in kg) of flour, water, and yeast required to make one batch of each bread type. The bakery receives an order for 5 batches of sourdough, 3 batches of rye, and 4 batches of wholemeal. Use matrix multiplication to calculate the total amount of each ingredient required to complete the order.