# Regarding transpose of matrix products

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In summary, the conversation revolves around a formula that was written on the board in a Linear Algebra class. The formula contains matrices and the speaker is questioning why certain matrices have specific properties. The expert suggests that the formula is incorrect and provides the correct version. The speaker also mentions difficulties with the professor's teaching style and asks for tips on how to format math equations.
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Starting out a Lin Alg class - my prof wrote this on the board.

(ABC-1Dt)t = DC-1BtAt

On the right hand side, I get why D is D, why A and B are now both transpose, but why is C still inverse? I know the rule (D-1)t = (Dt)-1, but I do not see how the heck it applies here or what would make the original equality true.

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It's incorrect, it should be

$$(ABC^{-1}D^t)^t = D (C^t)^{-1} B^t A^t$$

Ugh, I KNEW it was incorrect. He wrote it on the board right before we took a test - he was answering questions - so despite knowing the write answer as you put it, I wrote the wrong thing on the exam. I would have asked, but out the tests came.

This might have something to do with his being an octogenarian. Half hour late to his office hours too and forgets who the heck you are. Son of a...

Thanks for your help. Also, how do you make the script look so nice for writing math?

FWIW, some people write ##C^{-T}## instead of ##(C^{-1})^T## or ##(C^T)^{-1}##

Maybe you misread ##C^{-T}## as ##C^{-1}##.

AlephZero said:
FWIW, some people write ##C^{-T}## instead of ##(C^{-1})^T## or ##(C^T)^{-1}##

Maybe you misread ##C^{-T}## as ##C^{-1}##.

Could be - wouldn't be in keeping with his writing style though. And it wouldn't be the first mistake we've dealt with.

Thanks a bunch!

## What is the definition of the transpose of a matrix product?

The transpose of a matrix product is the product of the transposes of the individual matrices, performed in reverse order. This means that if A and B are matrices, then the transpose of their product AB is equal to the transpose of B multiplied by the transpose of A (i.e. (AB)^T = B^T * A^T).

## What are the properties of the transpose of matrix products?

The transpose of matrix products have several properties that make them useful in matrix algebra. These include the fact that the transpose of a sum is equal to the sum of the transposes (i.e. (A+B)^T = A^T + B^T), and that the transpose of a scalar multiple is equal to the scalar multiple of the transpose (i.e. (kA)^T = kA^T).

## How is the transpose of a matrix product useful in solving systems of equations?

The transpose of a matrix product is useful in solving systems of equations because it allows for the use of the transpose-inverse method. This method involves taking the transpose of a matrix equation, and then multiplying both sides by the inverse of the resulting matrix. This can help to simplify and solve complex systems of equations.

## What is the relationship between the transpose of a matrix product and the identity matrix?

The transpose of a matrix product is related to the identity matrix in that the product of a matrix and its transpose will always result in the identity matrix. In other words, if A is a matrix, then AA^T = I, where I is the identity matrix.

## How can the transpose of a matrix product be used in data analysis?

The transpose of a matrix product can be used in data analysis to manipulate and transform data. For example, in statistics, the transpose of a matrix product can be used to calculate the correlation between variables. It can also be used to perform operations such as principal component analysis and data compression.

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