Compute (B^T)(C) and multiply the result by A on the right

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In summary, the conversation discusses computing the product of three matrices A, B, and C, specifically (AB^T)C, (B^T)C multiplied by A on the right, and the equivalence of (AB^T)C and ((B^T)C)A. It is clarified that "multiply the result by A on the right" means (B^T)C multiplied by A, not the other way around. It is also noted that matrix multiplication is not commutative, but it is transitive.
  • #1
nietzsche
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"Compute (B^T)(C) and multiply the result by A on the right"

Homework Statement



I have three 3x1 matrices A, B, and C.

First I have to multiply (AB^T)C. I understand the result is a 3x1 matrix (I think).

Next the question says "Compute (B^T)C and multiply the result by A on the right. (Hint: (B^T)C is 1x1).

The last question is "Explain why (AB^T)C = ((B^T)C)A.

Homework Equations





The Attempt at a Solution



I don't understand the second question. Does "multiply the result by A on the right" mean ((B^T)C)A or A((B^T)C)? Because I find that the former is not defined but the latter is.

But then I got confused because of the last question.

Can anyone clarify this for me? I'd appreciate it, thanks.
 
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  • #2


Multiply X on the right by Y means XY, not YX. As you have discovered, matrix multiplication is not commutative. It is transitive, however.
 
  • #3


D H said:
Multiply X on the right by Y means XY, not YX. As you have discovered, matrix multiplication is not commutative. It is transitive, however.

Thanks a lot!
 

What does it mean to "Compute (B^T)(C) and multiply the result by A on the right"?

This means to calculate the transpose of matrix B, multiply it by matrix C, and then multiply the resulting matrix by matrix A. This is a common operation in linear algebra and is often used in scientific and mathematical calculations.

What is the purpose of computing (B^T)(C) and multiplying it by A on the right?

This operation is useful for transforming data or performing calculations on multiple matrices in a specific order. It can also be used to solve systems of linear equations and perform other mathematical operations.

What are the steps involved in computing (B^T)(C) and multiplying it by A on the right?

The first step is to transpose matrix B, which involves switching the rows and columns of the matrix. Then, the transposed B matrix is multiplied by matrix C. Finally, the resulting matrix is multiplied by matrix A on the right. Each step involves multiplying corresponding elements and summing the products.

What are the key properties of this operation?

One key property is that it is distributive, meaning that (B^T)(C) can also be written as (C)(B^T). Additionally, this operation is associative, meaning that the order of multiplication does not affect the result. It is also important to note that this operation is only defined if the number of columns in matrix B is equal to the number of rows in matrix C.

How is this operation used in scientific research?

This operation is often used in data analysis and modeling, as well as in solving equations and performing calculations in fields such as physics, engineering, and economics. It is also essential in machine learning and other applications of linear algebra in computer science.

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