# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.7

7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Students should understand that sometimes, the planets align and the cosmos paves the way to their happiness. They might find \$5 in the pocket of their jeans or get asked out by the person they've been crushing on for months. Or sometimes, the math lesson for that day is incredibly easy.

Tell your students to check their back pockets, because this may turn out to be one of those days.

Students should understand that when a matrix is multiplied by a constant (called a "scalar"), each entry in the matrix is multiplied by that constant. Sounds like the distributive property… again. At this rate, we'll run out of things to distribute.

To use a concrete example, let's pretend you own a fleet of cars. We might as well dream big, right? You make a matrix to illustrate the price of keeping them all gassed up (no fart jokes, please).

Much to your dismay (and everyone else's, for that matter), you find that the price of gas has gone up 23% over the past few months. What do you suppose happens to the price of gassing up each of those cars? They're all multiplied by 1.23 (1.00 to keep the price the same, and the 0.23 because of the price increase).

Easiest math lesson ever, right? Now, they'll have to wait until class is over, but you can tell your students to check their texts because there may be a message from that special someone waiting for them.

#### Drills

1. Which property comes closest to describing scalar-matrix multiplication?

Distributive

When a term is outside parentheses, it multiplies each of the terms within the parentheses. Scalar multiplication is kind of the same thing. It multiplies each of the terms within the matrix.

2. Find the product.

Multiplying a matrix by a scalar is the same as multiplying every number in the matrix by that scalar. That means we have to multiply 2, 3, -1, and 4 by 2, which gives us 4, 6, -2, and 8. The matrix with those numbers in the proper positions is (C).

3. Find the product.

Fractions are constants too, so the same process for multiplying scalars and matrices holds. Since our particular scalar is negative, that switches the sign of every entry, too. All we really need to do is find the first number, and we narrow it down to (A) instantly.

4. A matrix is multiplied by a scalar of ½. If the resulting matrix is given below, what was the original matrix?

If we're given the produced matrix and the scalar, we can divide that matrix by the scalar to calculate the original one. In other words, we multiply the produced matrix by the reciprocal of the scalar. In our case, that means multiply each entry we're given by 2. That'll get us (B) as the right answer.

5. If the matrix is multiplied by a scalar to form the matrix , what is the value of the scalar?

-3

Each term in the product equals -3 times the corresponding term of the original matrix. We can find this by dividing each term of the resulting matrix (or just one, really) by the corresponding term(s) in the original matrix. Let's hope Keanu Reeves is okay with all this matrix meddling, though.

6. If the matrix results from multiplying a scalar and , what is the value of the scalar?

Such a scalar does not exist

How can we multiply 2 by a number and get -4, and multiply 1 by the same number and get 2? It's impossible, because we can't pick and choose when we want to add the negative sign and when we don't. So we can't pick a scalar as our answer, so it doesn't exist.

7. Beth has decided to exercise more. She's made a matrix of the number of minutes she spent working out for the past week. She figures that an increase of 10% would be healthy. If her original matrix is . Which of the following represents the product of her planned exercise matrix?

Since she wants to increase each session by 10%, she wants to multiply each by 1.1. That gives us (C) as the right answer. (Why not multiply by 0.1, you ask? Because that would only give the amount of the increase and not the total time. Did you really think a 5-minute workout was healthier than a 50-minute workout?)

8. The prices of iTunes music will go up by 5% soon. We'll still pay, though, because illegally downloading music is bad. If the current prices of songs or albums we want to buy are  in dollars, by how much will each item increase in price?

The question asks for the increase in price, not the total price, so we need to multiply the matrix we're given by 0.05. If we multiply each entry in the matrix by that amount, we should get (C) as our answer. All in all, that's only a \$1.80 increase in total. Things could be worse.

9. If the high and low temperatures for a 5-day forecast is given by the following matrix. If the temperatures were really 7% cooler on all those days, what was the actual forecast? (We'll assume these degrees are in Fahrenheit, otherwise we'll need it to be a lot more than 7% cooler.)

The top row represents the higher temperatures and the second represents the lower ones. That means we can automatically forget (B) because they're switched. If the temperatures are 7% cooler, that means they need to be multiplied by 1 – 0.07, or 0.93. If we make all the temperatures cooler, we'll get (A) as the answer. We'll also want to put on a sweatshirt, most likely.

10. Is there any relationship between the size of the original matrix and the size of that matrix after being multiplied by a scalar?