How to find the determinant of two non-square matrices?

Question

I have to find the determinant of a non-square matrix. But because we can have the determinant of square matrices only. Then, how can I transform a $2 \times 3 $ matrix to be a square one?

EDIT: I have two matrices, the first one of size $2 \times 3 $ and the second one of size $3 \times 2 $, I want to find the determinant of their product without finding their product.

You can pad it with zeros but the determinant is zero. Give us more context to try to help you better. — Sean Roberson, Mar 13 '18 at 07:35
Welcome to MSE. Please read this text about how to ask a good question. — José Carlos Santos, Mar 13 '18 at 07:42
Why do you want to find the determinant of a non-square matrix? — Siong Thye Goh, Mar 13 '18 at 07:43
The short answer is what you yourself already said: "We can have the determinant of square matrices only." Any "transformation" of your original matrix into a square matrix will allow you to take the determinant of the transformed matrix. This however will not be the determinant of the original nonsquare matrix. Further, there are infinitely many different "transformations" you can make with the choice of which one to use being entirely subjective.. — JMoravitz, Mar 13 '18 at 07:51
The only interesting (in general nonzero) determinant that can be associated to a $3 \times 2$ matrix $A$ is $\det(A^TA)$ where $A^TA$ is $2 \times 2$ matrix. — Jean Marie, Mar 13 '18 at 08:08
@SeanRoberson I knew that and tried it. I edited the question. — Learning Equals Success, Mar 13 '18 at 10:57
@JMoravitz I read that question before and didn't find any useful answer in my point of view. That's why I asked a new question. I know that but it's also possible to have the determinant of a nonsquare matrix. Sorry; I wasn't very clear in my question before. But if I found the determinant of the two matrices then the problem is solved. — Learning Equals Success, Mar 13 '18 at 11:01

score 0 · Answer 1 · answered Mar 13 '18 at 11:05

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Luckily, I asked my professor and it appeared that it can be solved by Binet-Cauchy Formula.

You can read about it here: https://sites.math.washington.edu/~morrow/335_12/CauchyBinet1.pdf

answered Mar 13 '18 at 11:05

Learning Equals Success

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1

This solves the problem in your edit, but 1) it doesn't solve the problem in the title, and 2) it's not clear to me that this is any easier than computing the product of the two matrices and then computing the determinant. – Gerry Myerson Mar 13 '18 at 11:45
@GerryMyerson I agree with you that the title doesn't relate to the question, so I edited that too. As for your second point, if the question or the exercise doesn't want you to find their product, then you have no choice but to use that formula. – Learning Equals Success Mar 13 '18 at 12:28

How to find the determinant of two non-square matrices?

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