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### Course: Precalculus > Unit 7

Lesson 11: Properties of matrix multiplication- Defined matrix operations
- Matrix multiplication dimensions
- Intro to identity matrix
- Intro to identity matrices
- Dimensions of identity matrix
- Is matrix multiplication commutative?
- Associative property of matrix multiplication
- Zero matrix & matrix multiplication
- Properties of matrix multiplication
- Using properties of matrix operations
- Using identity & zero matrices

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# Using properties of matrix operations

Sal determines which of a few optional matrix expressions is equivalent to the matrix expression A*B*C. This is done using what we know about the properties of matrix addition and multiplication. Created by Sal Khan.

## Want to join the conversation?

- Don't we get into problems here as A^2 has the dimensions of A * A, and A * A might not be defined (2x3 * 2x3)? Can we do algebraic manipulation even if we don't yet know if the operation is defined for all matrices?(10 votes)
- The question states that all the matrices are square, so it isn't a problem here.(41 votes)

- How do we know that A(BC+A) is equal to ABC + A^2 and not BCA + A^2 or BAC + A^2?(7 votes)
- Matrix multiplication is not (in general) commutative for multiplication. That is, if you change the order of multiplication (AB to BA) you don't always get the same answer. (In fact, sometimes, because of the dimensions of the matrices, you cannot even find the reversed product.) So you can't treat these A's, B's, and C's the same as you did in Algebra I because they are representing matrices (not commutative) and not representing real numbers (commutative). You must keep the A, B, and C in the same order.(15 votes)

- 3:10How do you know that ABC+A^2 is defined? They could have diferent dimensions(4 votes)
- We're told that A, B, and C are square matrices. We also have that ABC is defined. So A, B, and C must all be the same dimension (if they weren't, one of those multiplications would be undefined).

When we multiply square matrices of equal dimension, we get another matrix of the same dimension. So ABC must be the same dimension as A, and A^2 must be the same dimension as A.

So ABC and A^2 must have the same dimension.(8 votes)

- at2:50sal talks about a "zero matrix" what does he mean by it?(2 votes)
- A
*zero matrix*is a matrix all of whose entries are zero.(11 votes)

- Is this a reference to the sci-fi series Ender's Game? I noticed the names are the same ones used in the book, however Commander Graff is addressed colonel not commander.(7 votes)
- For the very first question (battle school), why does it have to state that ABC are square matricies? Does that affect the answer in any way and if so how?(5 votes)
- It matters a lot. For example,if a is a 2*3 matrix A * A might not be defined (2x3 * 2x3) because there is a different no. of columns in the first and rows in the second. But if it is a square matrix, then it will always be definable.(4 votes)

- Wouldn't the parenthesis force the addition of the ABC clause to the AA clause prior to subtracting the AA clause? I am asking because I do not know if the same Order of Operations applies with matrices.(5 votes)
- well u can simplify an algebra problem first before order of operations right(1 vote)

- how can we square a matrix ?,like A^2 ?(4 votes)
- Squaring something (like a matrix or a real number) simply means multiplying it by itself one time: A^2 is simply A x A. So to square a matrix, we simply use the rules of matrix multiplication. (Supposing, of course, that A can be multiplied by itself: not all matrices can be multiplied. So my question to you would be: What kind of matrix can be squared?)(2 votes)

- Just for clarification, that is a Enders game reference, if I'm correct?(4 votes)
- What's a square matrix?(2 votes)
- It's a matrix that has the same number of rows and columns.(4 votes)

## Video transcript

Voiceover:In order to
get into Battle School cadets have to pass a
rigorous entrance exam which includes mathematics. Help Commander Graff grade the next wave of students' tests. The last step of a problem in the matrix multiplication section is the matrix A times B times C where A, B, and C are square matrices. Which of the following
candidates have answers that are equivalent to this expression? Select all who are right
for any A, B, and C. Select all that apply. I encourage you to pause this
video and think about it. Which of these expressions
for any square matrices A, B, and C are equivalent to this right over here. I'm assuming you've given a go at it. Let's think through each of them. This one is B, A, C so if they have changed the order and we've already seen
that matrix multiplication is not commutative in general and so this will not be
true for any square set of matrices A, B, and C, so this is not going to be true. Matrix multiplication is not commutative. Here you have Bernard, who
says A times, C times B. We already know that that's going to be the equivalent to A, C, B which once again they've swapped the order between the B and the C, matrix multiplication is not commutative. You can't just swap order and expect to get the same product for any square matrices A, B, and C so we could rule that one out. A times, B, C, so we've already seen
matrix multiplication is associative, so this is the same thing
as A times B, times C which of course is the
same thing as A, B, C. What Caren has right
over here, that is right, that is equivalent for any
square matrices A, B, and C that is equivalent to A, B, C. Now Ducheval, let's see, now this looks like a
bit of a crazy expression but let's think it through a little bit. First of all matrix multiplication, as long as you keep the order right, the distributive property does hold. This first part right over
here is equivalent to ... Let me write this down,
this one's interesting. We have A times B, C
plus A, minus A squared. You can actually distribute this A and I encourage you to
prove it for yourself maybe using some two by two
matrices for simplicity. This is going to be equal to, this part over here is going to be A, B, C plus A A, A times A which
we could write as A squared and then we're going
to subtract A squared. These two things are going to cancel out, they're going to end up
being the zero matrix and if you take the zero, so these are going to be the
zero matrix right over here. If you take the zero matrix
and add it to A, B, C you're just going to end up with A, B, C. This one was a little bit tricky, this one actually is equivalent. This one is right and this one is right. Here this is A times B, plus C so this is kind of not-y right over here. They're not even multiplying B and C, so this one's definitely
not going to be true for all square matrices A, B, and C.