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# Associative property of matrix multiplication

Sal shows that matrix multiplication is associative. Mathematically, this means that for any three matrices A, B, and C, (A*B)*C=A*(B*C). Created by Sal Khan.

## Want to join the conversation?

• Is This property true for more than 3 matrices? e.g. ABCD = (AB)(CD) or (A(BC))D or A(B(CD)) ?
• Yes, this property is true for more than 3 matrices!
• There are so much variables, is there any way to keep track of them all?
• Sometimes it helps to use one letter for each matrix, and use subscripts to denote the different entries in each. For example instead of going up to l use a1,a2,a3,a4, for the first matrix, b1,b2,b3,b4 for the next, and c1,c2,c3,c4, for the last. this allows you to better see what matrix each entry came from.
• In the end, you explained that iae = aei, but I still don't understand how? Then, you mention scalar property, which only make me more confuse, how is this related to scalar multiplication?
• Each of the entries within a matrix is a scalar. By now you are assumed to realize that when you multiply (2*3)*4, for instance, you will get the same thing as when you multiply (3*4)*2. The associative and commutative properties of scalar multiplication are well-established and familiar, but you might not have called them that.
• is this the only way to multiply three or more matrices?
• No, kinda. When you see ABC, it is implied that what you are actually doing is this: (AB)C (assuming you follow the rules of PEMDAS). This video shows that the operation ABC is associative, meaning ABC not only implies you are doing (AB)C, but you could also do A(BC). Remember, as long as you multiply the matrices in order, (matrix multiplication isn't commutative) you don't have to worry about parenthesis placement as such presented in this video.
• I get the proof but I struggle to understand it intuitively. My brain doesn't let me generalize the proof for non-square matrices without checking the specific cases.
Any help?
• Matrices represent a certain type of transformations of space. An nxm matrix turns an m-dimensional space into an n-dimensional space, and multiplying matrices corresponds to applying one transformation after another.

Therefore, matrix multiplication is a specific type of function composition. Because function composition is associative, so too is matrix multiplication.
• If there are three matrices namely A,B and C with their order (3*2),(2*4) and (4*1) respectively. Is it possible to apply Associative property i.e. A.(B*C)=(A*B).C in this context.
• Yes, that is correct. The associative property of matrices applies regardless of the dimensions of the matrix.

In the case `A·(B·C)`, first you multiply `B·C`, and end up with a 2⨉1 matrix, and then you multiply `A` by this matrix.

In the case of `(A·B)·C`, first you multiply `A·B` and end up with a 3⨉4 matrix that you can then multiply by `C`.

At the end you will have the same 3⨉1 matrix .
• Does this only work for square matrices?
• Check this and you will see it works as well:

|a| * |c d| * |e|
|b|________|f|
(1 vote)
• Is there a way to generically show it for n*n matrices?