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.