# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.11

11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Students should feel pretty comfortable with matrices by the time you get to this standard. In particular, we hope they're feeling good about matrix multiplication, because the worlds of matrices and vectors are about to collide. Hopefully, they'll emerge vectorious. (Get it?)

Students should understand that a vector can be represented by a one-column matrix. Each number represents the components of the vector. (This can happen in 2D space or in 3D space.)

Let's say we have a vector with components <4, 1, 2> (meaning it moves 4 units along the x-axis, 1 unit along the y-axis, and 2 units along the z-axis). We can represent it as a matrix with one column:

Easy enough, yes?

Students should understand how to multiply a matrix by a vector. So let's take a matrix of appropriate dimensions and multiply it by our vector. (Notice the order in which we did that. The order matters.)

Remind your students and remind them again: we want to go across the rows of the matrix as we work our way down the vector.

To start with, we multiply 2 × 4 + 3 × 1 + 2 × 2 = 8 + 3 + 4 = 15.

For the second row, we have 1 × 4 + 5 × 1 + 3 × 2 = 4 + 5 + 6 = 15.

And finally, our last row gives us 0 × 4 + 3 × 1 + 6 × 2 = 0 + 3 + 12 = 15.

Our resulting vector is

Students should understand that multiplying a matrix and a vector is really just a combination of multiplication and addition—a way to transform that vector in 2D or 3D space. When we multiply a matrix and a vector together, we transform that vector in some way and change its components.

#### Drills

1. What is the product of the following matrices?

There is no solution

In order to multiply matrices, we need to have the same number of columns in the first matrix as rows in the second. Here, since the first matrix has 3 columns and the second matrix has 1 row only, we can't multiply them together. That's it. The end. Bye-bye.

2. How can we represent a vector with components of <2, 7> as a matrix?

We represent vectors as matrices by sticking all the components in the same column and giving each one its own row. We don't need extra zeros all around either, since we won't ever need to fill them up.

3. How can we represent a vector with components of <-1, 2, 9> as a matrix?

Just because a vector has three instead of two components doesn't mean the rules for representing them change. Instead of two rows, we'll just have three. One column, three rows. That's all.

4. What is the product of the following matrices?

[35]

We go across the matrix as we go down the vector: 4 × 2 + 9 × 3 = 8 + 27 = 35. It's okay that the resulting matrix is just one number. Sometimes that happens. It's not true in all cases, but here, size doesn't matter.

5. Which of the following matrices can we multiply by a vector with components <1, 1, 1>?

A vector with components of <1, 1, 1> can be represented by . Since it is the second matrix in the operation, we need a matrix that has the same number of columns as rows in this matrix. That means any matrix with three columns will do. The only one in our options is (B).

6. What is the product of the following matrices?

First, we multiply the top row and the vector and add them all up: 3 × 4 + 2 × 2 + -1 × 3 = 12 + 4 – 3 = 13. Then, the second row and the vector gives us 6 × 4 + 1 × 2 + 0 × 3 = 24 + 2 + 0 = 26. Since we have two rows in the first matrix and one column for the vector, our resulting matrix also has two rows and one column.

7. What is the product of the following matrices?

There is no solution

Matrix multiplication is not commutative, which means that the order matters. (Tell your students to repeat that out loud. Twice.) In order to multiply matrices, you need the same number of columns of the first matrix as rows of the second, not the reverse. As hard as we try, there's just no way to multiply these matrices together.

8. The product of the following matrix and a vector will result in a vector with components of <24, 71>. What are the components of the vector?

<9, 0, 2>

Since the first matrix involved in multiplication has three columns, we need our vector to have three components. Otherwise, we won't be able to multiply them. If we try both (A) and (B), we'll see that the only one that works is (B).

9. The following matrix and a vector with components <-1, 7, 2> are multiplied together. What is the result?

We know that the first matrix has three rows and the second matrix (the vector) has one column. That means our resulting matrix will have three rows and one column, so that's either (A) or (C). As usual, add the products of the rows and column (0 × -1 + 7 × 6 + 2 × 2 = 46, then -1 × -1 + 2 × 7 + 5 × 2 = 25, and finally, 8 × -1 + 2 × 7 + 3 × 2 = 12). That looks like (C) to us.

10. Which of the following matrices will transform a vector with components <1, 2, 3> to a vector with components <10, 11, 12>?

We want a matrix multiplied by  to be . First, we know that it has to have three columns (for multiplication to be possible) and three rows (so that the resulting matrix has three rows, too). It can't be (B), because that's the identity matrix and will give us the same vector again. If we just go through the actual multiplication, we'll see that (A) gives us what we want.