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

Lesson 9: Properties of matrix multiplication

# 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?
• The question states that all the matrices are square, so it isn't a problem here.
• How do we know that A(BC+A) is equal to ABC + A^2 and not BCA + A^2 or BAC + A^2?
• 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.
• How do you know that ABC+A^2 is defined? They could have diferent dimensions
• 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.
• at sal talks about a "zero matrix" what does he mean by it?
• A zero matrix is a matrix all of whose entries are zero.
• 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?
• 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.
• 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.
• 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.
• well u can simplify an algebra problem first before order of operations right
(1 vote)
• how can we square a matrix ?,like A^2 ?
• 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?)