Sample Question

Explain with examples how each of the following properties are or are not true with matrices.

1. Commutative property of addition.
2. Commutative property of multiplication.

What other properties do you know about that work with real numbers?
Explain how you think it does or doesn’t hold true for matrices.

Sample Response

Let A = , B = , and C = .

1. A + B will be the same as B + A since you just add each part of the matrices and it doesn't matter with real numbers if you add in different order. Take a look:
   
   A + B = =
   B + A = = .
   
   It wouldn't have mattered what the entries were in our matrices, the order in which we add them would still be fine since we're just adding entries.
   So the commutative property of addition is true for matrices.
   
   
2. AB will not be the same as BA. Take a look:
   AB = =
   BA = =
   So the commutative property of multiplication is not true for matrices.
   
   

I know about the associative property of addition. (5 + 6) + 7 = 5 + (6 + 7).
Since adding matrices is really just adding each corresponding entry (like I showed in #1), then the associative property of addition will be true for matrices.
Take a look:
(A + B) + C = =
A + (B + C) = =
The associative property of addition is true for matrices.

I also know about the distributive property. 4(6 + 2) = 4*6 + 4*2.
The distributive property will be true also for matrices. Take a look:
3(A + B) =
3A + 3B =
Since the multiplying here is really only making each entry bigger, then you have regular addition and it's not going to be a problem. It'll work for any numbers.
The distributive property is true for matrices.