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## Algebra (all content)

### Course: Algebra (all content)>Unit 20

Lesson 5: Adding & subtracting matrices

Sal defines what it means to add or subtract matrices. He shows a few examples and discusses some important properties of matrix addition and subtraction. Created by Sal Khan.

## Want to join the conversation?

• Was the intuition of the mathematicians who defined matrix addition really that arbitrary? Can matrices be considered as vectors, in a way, with the addition and subtraction properties of matrices being similar to that of vectors?
• I guess Sal was being kind to say so and saved the better stuff at linear algebra courses where we learn that indeed matrices are group of vectors which are indeed at heart of the whole matter. We use matrix notation to solve n dimensional vector space linear systems.
• What are matrices used for? What is the point of a matrix?
• Is it possible to add a real number to a matrix, or is it undefined?
• Adding a scalar (or number) to a matrix is not defined. You can multiply a number by a matrix.
• In the video Sal keeps mentioning the mathematic mainstream, so does that mean, and who are they?
• Popular Mathematicians like Newton. Who influenced the world.
• Questions for +/- matrices of different dimensions. What if we add 0's to make matrices in even dimensions? Is there some reason we can't do this?
Ex
A [1 2 3] + B[ 1 2] = undefined
But what if we do this:
A [1 2 3] + B[ 1 2 0]= C[2 4 3]
• It's certainly true that [1 2 3]+[1 2 0] is well-defined, but [1 2 0] is not the same object as [1 2].

Besides, if we try to extend matrices like this, how do we know whether [1 2] should become [1 2 0] or [0 1 2]?
• At about , Sal said that you can't just put a matrix in any order when multiplying and dividing. I understand why you wouldn't be able to put the matrices in any order while dividing, but since multiplying is simply repeated addition, wouldn't the order of two matrices not matter?
• Actually, repeated addition of a matrix would be called scalar multiplication. For example, adding a matrix to itself 5 times would be the same as multiplying each element by 5.

On the other hand, multiplying one matrix by another matrix is not the same as simply multiplying the corresponding elements. Check out the video on matrix multiplication. Indeed, matrix multiplication is not commutative.
• so i can add or subtract matrices if they have the same dimensions (rows and columns), right?!
• Yes, a matrix can only be added or subtracted if they have the same dimensions.
• What are the point of matrixes? I don't really get how you would use this in real life.
• Half the things in math you won't really need in real life unless your going to be in a profession that is based on math or immensely involves it.
• At , Sal says that addition and subtraction with matrices of different dimensions is undefined. Why can't one just insert entries of 0s in the missing areas of a matrix in addition and subtraction? Would the matrix be different at that point? If so, is that the case with augmented matrices?
• Matrices as I understand were made for matrix multiplication, specifically with matrices with dimension nx1 (where n is any number) called a vector. So adding 0s does make them different. Some places moreso than others, but in the worst case it can make it so the two matrices can't be multiplied.

Augmented matrices are a special case, which is why they got their name. Specifically it is when a sum or difference of vectors, or more commonly a system of equations, actually has answers. the answers are not really part of the matrix so they get their own part of aan augmented matrix.

in non system cases, think about if you have just y = mx+b or y equals a number. if you have a number instead of y (and m of course) you can solve for x, while having y there makes it only so you can find a function you can graph.

Let me know if this didn't help