Using matrices to manipulate data: Game show | Matrices | Precalculus | Khan Academy

We're told in the beginning of each episode of a certain game show. Each contestant picks a certain door out of three doors. Then the game show host randomly picks one of the two prize bundles. After each round, each contestant receives a prize based on the door they picked and the bundle the host picked.

Matrix A represents the possible prizes for the first round. Alright, so for example, if the contestant picks door 3 and the host picks bundle 1, the prize is 300. But if the contestant picks door 3 and the host picks bundle 2, the prize would be zero dollars. All right. And then they say matrix B represents the possible prizes for the second round.

Alright, that's fair. They also tell us the second round can also be a lightning round. In this case, the prizes are doubled. Matrix C represents the possible prizes during a lightning round. Complete matrix C. So pause this video and see if you can figure that out, and then we'll work through this together.

Alright, so matrix C is a scenario where we're dealing with a lightning round. And remember, in a lightning round, the prizes are doubled, but it's the second round. It's doubled relative to what it would have been in the second round. So what it would have been in the second round is matrix B.

So another way to think about it is matrix C is going to be equal to 2 times matrix B. And we know when we multiply a matrix times a scalar like this times just a number, we just multiply each of these entries by that number. So let's do that. If we take 600 and multiply that by 2, that is going to be 1200.

And that makes sense. We just said for each corresponding scenario, the prizes are doubled in a lightning round. So if the contestant picks door one and the host picks bundle one, instead of six hundred dollars, it's going to be twelve hundred dollars. Keep going. Instead of two hundred dollars, it's going to be four hundred dollars.

All I'm doing here is I'm multiplying each of these entries by two to get the corresponding entry in matrix E. Keep going. Instead of three hundred dollars here, multiply that by two; you're going to get six hundred dollars. Instead of three hundred dollars here, you're going to get six hundred dollars. We're almost there.

Instead of zero dollars here, well, zero times two is still zero dollars. And then last but not least, instead of four hundred dollars right over here, that times two is going to be 800. Now there's one more question that they have below the screen right over here. Let me scroll up a little bit.

So they tell us matrix D is defined as follows: D is equal to A plus B. What does matrix D represent? So pause the video and think about that for a second. Well, if we add two matrices, we're going to add all the corresponding entries.

And so what it tells you is what is the combined prize for both rounds one and two based on what the contestant picks and what the host picks. So matrix D, that top left entry will tell you, okay, in total, if the contestant picked door number one and the host picked bundle number one, what would you get? Because it would be a hundred dollars plus six hundred dollars, so it would be total for rounds one and two, assuming we don't have the lightning round like we had in matrix E.