# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.9

9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Students should search their math memory banks for terms like "associative," "distributive," and "commutative." If they don't know those, we're in some serious trouble.

Remind them of their times tables. (They do still remember those, don't they?) Way back when, they probably worked really hard on the lower numbers—the 2's, 3's, 4's and 6's (come on, the 5's were pretty much freebies). Then they got to the upper numbers, and things got harder for some reason. But the first half of the table was easy, so they didn't have to memorize 7 times 3, because they already knew that 3 times 7 was 21.

That's the commutative property.

Well, we have some bad news for you. While multiplication of numbers is commutative, it turns out that multiplication of matrices is not. What that means in English is that the order of multiplication matters. A × B does not usually equal B × A. And that's assuming that we could even perform both multiplications.

Students should know that you're not making this up just to mess with them. That's really how math works. But they'll be relieved to learn that both the associative and distributive properties are still alive and well with matrices.

The associative property says that, as long as you keep the order the same, you can move parentheses around. That means A + (B + C) = (A + B) + C. In case they don't remember the distributive property, it goes like this: a(A + B) = aA + aB.

#### Drills

1. Which of the following properties do not apply to matrices?

Commutative

The commutative property states that a × b = b × a. This is true for numbers, but the multiplication process for matrices renders this property untrue when two matrices are multiplied. While it might sometimes be true for specific matrices, it cannot be regarded as a property that matrices have.

2. Why is it logical that the commutative property doesn't apply to matrix multiplication?

Because you multiply the row of the first matrix with the column of the second; the order matters

When we multiply matrices, we go across each row of the first matrix as we move down the column of the second. Switch the matrices around and you're working with the numbers in a different order. That means the order in which we multiply the matrices matters.

3. Why do the matrices need to be square if we're going to consider doing both A × B and B × A?

We want to go across rows and down columns, then switch them around. That will only work if the numbers of rows equals the numbers of columns.

Again, look at how we do matrix multiplication. We need to have the same numbers of rows as columns if we plan to reverse the order of the matrices. The rules don't change just because we switch the matrices around.

4. Is it ever possible for the product of two matrices to yield the same answer, regardless of the order in which they're multiplied?

Yes, since there are different combinations of numbers that will give the same entry in a matrix

The lack of a commutative property of multiplication for matrices simply means that you can't count on it working, not that it will never work. For example, of one of the matrices is a square matrix containing all 0's, the two answers will always be the same, regardless of order.

5. If matrix multiplication is not commutative, does that mean you cannot square the matrix?

No, but only square matrices can be squared, since the number of rows needs to equal the number of columns

Here's an easy way to remember it: we can only square a square matrix. To multiply one matrix by another, we need to go across the rows of the first and go down the columns of the second. If we're not going to run out of entries anything—which means the matrices can't be multiplied—we'll need a matrix where the number of rows equals the number of columns. That's what a square matrix is.

6. Explain why the distributive property works for matrices.

The distributive property with matrices is a combination of scalar multiplication and matrix addition, both of which work with matrices

A recap of the distributive property might be helpful. That's just a(A + B) = aA + aB. In this case, a is just a number (a scalar) and A and B are matrices. This property does work for matrices, and it's because of (B).

7. If the commutative property doesn't apply to matrices, why does the associative property apply?

The associative property has to do with rearranging parenthesis rather than the order in which terms are multiplied

The associative property states that A + (B + C) is the same as (A + B) + C. As answer (C) states, it has to do with the placement of parenthesis and not the order of operations (or surgeries) necessarily. While the order matters in matrix multiplication (thanks for nothing, commutative property), parentheses don't.

8. Which matrix, when multiplied with , will yield the same result regardless of the order in which they're multiplied?

If each of these matrices are multiplied together in both ways (which takes a while to do), then we'll see that the only matrix that works is (C). That's because it's a square matrix all zeros. That makes everything zero, regardless of the order.

9. Which matrix, when multiplied with , will yield the same result regardless of the order in which they're multiplied?

Why (D)? Well it's obvious, isn't it? They're the same exact matrix. It doesn't matter which is first and which is second because they're guaranteed to give you the same exact answer. (We're not saying it'll be fun to calculate, though.)

10. Which is an example of two matrices satisfying the associative and distributive properties? Let a be a scalar, and A, B, and C be three unique matrices.