About this video
A game show stores prizes in matrices, with rows for the contestant's door and columns for the host's prize bundle. In a lightning round the second-round prizes are doubled, so the lightning-round matrix is C = 2B, found by multiplying every entry by 2. Adding A + B gives D, the combined prize for rounds one and two.
You'll be able to
- read what an entry in a prize matrix means.
- use scalar multiplication to scale every entry of a matrix in context.
- explain what the sum of two prize matrices represents.
Key words
- scalar
- A plain number, like 2, that can multiply a matrix.
- scalar multiplication
- Multiplying every entry of a matrix by the same number.
- matrix addition
- Adding the entries in matching positions of two same-sized matrices.
Transcript
- [Instructor] 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. All right, so for example, if the contestant picks door three and the host picks bundle one, the prize is $300, but if the contestant picks door three and the host picks bundle two, the prize would be $0, all right.
And then they say matrix B represents the possible prizes for the second round. All right, 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. All right, 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 two 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 two, that is going to be $1,200. 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, host picks bundle one, instead of $600 it's going to be $1,200.
Keep going, instead of $200, it's going to be $400. All I'm doing here is I'm multiplying each of these entries by two to get the corresponding entry in matrix C. Keep going, instead of $300 here, multiply that by two, you're going to get $600. Instead of $300 here, you're going to get $600.
We're almost there. Instead of $0 here, well, zero times two is still $0. And then last but not least, instead of $400 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 pick bundle number one, what would you get? Cause it would be $100 dollars plus $600. So it would be total for rounds one and two, assuming we don't have the lightning round like we had in matrix C.
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