If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Linear algebra>Unit 2

Lesson 3: Transformations and matrix multiplication

# Distributive property of matrix products

Showing that matrix products exhibit the distributive property. Created by Sal Khan.

## Want to join the conversation?

• A x (B + C) works because the columns of A (m) equals the rows of B+C (m) but I don't understand how (B+C) x A works because the columns of B+C (m) doesn't equal the rows of A (k). Any ideas?
• Great question! I was wondering the same thing. I think it would have been better if he used three new matrices D, E and F for the second distributive proof, instead of recycling A, B and C.
• Its not dealt in this video, but I cant find a video that explains it. My doubt is: What happens with (A*B)^2 and with (A+B)^2 and (A-B)^2. I dont get how am I not to alter the order since we are dealing with matrices.
• Around , Sal uses the term "well-defined." Is there a difference between the term "defined" and the term "well-defined?"
• Cool question. (I chuckled a bit as I heard Sal say, "well-defined," three times in rapid succession, something I had not noticed when I watched the video the first time.)

The best I could come up with is that the term "well-defined" is used specifically for functions that have a specific output for a given input regardless of the form of the input (e.g. a fraction or decimal). https://en.wikipedia.org/wiki/Well-defined

The definition of the term, "defined" is either a silly question or a great question with a complicated answer. In my experience "defined" is more generally used for terms or concepts in mathematics (think geometric definitions) as opposed to specific functions. https://www.quora.com/What-is-the-definition-of-a-definition-in-mathematics (take a look at both answers, they seem contradictory and simultaneously both quite accurate.)

• Given matrices A,B, and C are all 2 x 2, determine whether the equatiom is true for the given matrices

C(A+B) = AC +BC