Today, we’ll explore matrix multiplication and how we can optimize it using an approach called dynamic programming. Let's first recall—when multiplying two matrices, what do we need to ensure?

Exactly! If we have a matrix A that's m by n and another matrix B that's n by p, the resulting matrix will be m by p. The actual computation involves multiplying corresponding elements in a row from A with a column from B and summing them up. How complex do you think this operation is?

Right! It takes O(mnp) time, but let's remember this as M = m * n * p. Awesome quick memory aid—just use the acronym M! Let's jot that down.

Let’s now dive into an interesting property of multiplication known as associativity. For example, if we multiply three matrices A, B, and C, does the order we multiply them in matter?

Correct! However, despite the result being the same, the efficiency can change drastically based on how we group the multiplications. Can you all think of how this could affect the number of calculations?

Exactly! This brings us to the core of optimizing the multiplication sequence. Keep that in mind as we analyze how to calculate these costs more systematically through dynamic programming.

Now, let’s refocus on our goal: minimizing the total multiplication costs. We want an efficient method to determine the best grouping of matrices. We’ll use dynamic programming. What do you think it means to break the problem down into smaller segments?

Great point! We can define a cost function and compute results for each segment recursively. Each time we compute, we check all possible splits of matrices to find the most efficient way. Can anyone recall how many sub-problems we’ve encountered before?

Exactly! That’s a crucial detail to keep in mind. Let’s apply this thought to our task of filling out the cost table. It’s a structured approach indeed.

Now that we have our recursive approach to solving sub-problems, let’s discuss how to fill the cost table systematically. What would our base case look like?

Perfect! So the cost of multiplying a single matrix is zero. As we fill in the cost table, we’ll move diagonally—filling one segment and using previously computed entries. Why do you think that makes sense?

Exactly! By starting from the basics and gradually working up, we minimize redundant calculations. In what time complexity do you think, should we expect this method to execute?

Spot on! Let's summarize what we learned today about matrix multiplication costs—its associative nature and how the sequence impacts efficiency.