# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.8

8. Add, subtract, and multiply matrices of appropriate dimensions.

That's pretty self-explanatory.

Both the addition and subtraction of matrices are incredibly easy. If your students ditched class the day you taught addition and subtraction of matrices, they'd probably be right if they just guessed how to do it. Naturally, they're responsible kids and would never dream of ditching a class as enthralling as yours.

The rule is just to add or subtract numbers that are in the same location. So the number in the top left corner is added to or subtracted from the number in the top left corner.

Students should understand one critical concept about the addition and subtraction of matrices of different sizes. It can't be done. Matrices need to have the same numbers of rows and columns to perform those particular operations.

Here are some examples.

Subtraction

As for multiplication, we should be straight with you. It's pretty tricky.

Here's how multiplication works: we move along the top row of the first matrix and down the second row of the second. Multiply the elements, and then add them up.

No, we aren't kidding. We aren't cruel enough to kid about something like this.

So before we even start, in order to be multiplied, the number of columns of the first matrix has to equal the number of rows of the second. Otherwise they can't be multiplied.

You might want to go easy on the students first, and start with square matrices.

First we move across the top row as we move down the first column. The answer goes in the top row, first column.

3 × 2 + 5 × 8 = 46

Now for the top row, second column.

3 × 6 + 5 × 9 = 63

Those are our two numbers for the top row. Now we'll move onto the bottom row of that first matrix, starting with the first column.

1 × 2 + 4 × 8 = 34

And finally, we'll calculate the bottom row, second column.

1 × 6 + 4 × 9 = 42

That means our resulting matrix is .

Students should know that it's not that bad once they get the hang of it. There are two tricks. The first is physically moving your fingers across and down the matrices so we know which numbers to multiply and add. The second is knowing where to put the answer. Just look at which row you multiplied by which column, and that will be where the answer goes.

Finally, students should know not only how to perform these functions, but how to ask for help if they need it. Matrices are friends, but they're not always easy to get along with.

Here is a recap video of multiplying matrices.

#### Drills

1. What is the sum of the following matrices?

Remember, addition is easy. You simply add the entries that occupy the same spots, and put the sums into that spot. Adding and subtracting aren't too crazy. It's multiplication that we have to watch out for.

2. Subtract.

Remember, we subtract entries in the same spot, and then write down what we get. So, our top row is 10 – 3 = 7, followed by 2 – 6 or -4. The bottom row is 4 – (-1), which is 5, then 6 – 0 which is 6. Our answer should always be the same size as the two matrices we started with, so (C) and (D) aren't right.

3. What is the sum of the following matrices?

If we're adding the two matrices, all we have to do is combine the numbers in the corresponding spots and that's our answer. We can do basic arithmetic (hopefully), so we'll end up with (C) for the answer.

4. Can the following matrices be multiplied together?

Yes, because the number of rows in the first matrix is equal to the number of columns in the second matrix

We told you multiplying matrices was complicated. Since multiplication of matrices happens by multiplying the rows by the columns, the number of rows in the first matrix has to match the number of columns in the second matrix.

5. Is the same as ?

No, because the rows are multiplied by the columns

We can do a quick calculation to double check. If we have , the first number is aw + by. If the matrices were reversed in order, the first number would be wa + cx. Since by doesn't necessarily have to equal cx, order does matter because the rows are multiplied by the columns. This is also a prelude to the next standard.

6. What is the product of the following matrices?

If we multiply the first row by the first column, we get 3 × 2 + 5 × -7 = -29. If we continue multiplying as such (or just stop there), we'll be able to tell that (C) is the right answer. See? It's not too complicated, right? Well, not yet, anyway.

7. What is the product of the following matrices?

First, we know it's possible to multiply these two matrices because the number of rows matches the number of columns of the second. If we multiply the first row by the first column, we will have 3 × 2 + 5 × 3 +6 × -6 = -15. The first row multiplied by the second column is 3 × -1 + 5 × 6 + 6 × 0 = 27. This is enough to point us to (B).

8. What is the solution to the following?

The square of the matrix is the same as . Multiplying the first row and the first column gives us -4 × -4 + 2 × 3 = 22. If we continued to multiply, we'd end up with (C) as the matrix.

9. What is the product of the following matrices?

It is possible to multiply these matrices together because the rows and columns match up. If we multiply 2 × 5 + 0 × 6 + 4 × 1 = 14. Then, the first row multiplied by the second column is 2 × 4 + 0 × 3 + 4 × -2 = 0. Since that was still in the first row, we know that each row in the resulting matrix needs two columns. The only matrix that satisfies all those is (B).

10. What is the solution to the following?