Prove that if A is nonsingular then the transpose of A is nonsingular.

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In summary, the conversation highlights the struggles of writing proofs after a long break and the need for assistance. The main focus is on finding the transpose of the product of two matrices and the identity (AB)^T = (B^T)(A^T). The person plans on seeking help from their professor, despite language barriers, and is reminded to check the accuracy of certain identities.
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seang
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I haven't written a proof in 8 years. Linear Algebra proofs are going to be the death of me. I honestly don't know where to begin. I read a sort of primer on proof writing, but I could use a human walk through or some help.

So far, I have:

there exists a B such that AB = BA = I.


then sometime later:

(A^T)B = B(A^T) = I

Beyond this I have no idea. I'm going to visit my prof tomorrow and see what he has to say. The trouble is, is that his english is fair to poor. A heads up or some honest help would be friendly,
Thanks.
 
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  • #2
What happens when you take the transpose of the product of two matrices. That is, what is (AB)^t ?
 
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  • #3
(AB)^T = (B^T)(A^T)

(A^T)^-1 = (A^-1)^T...you can switch the transpose and inverse.

Then (A^T)(A^T)^-1 = ...

Note: You should check those identities, I can't remember if they are correct.
 

Related to Prove that if A is nonsingular then the transpose of A is nonsingular.

1. What is the definition of a nonsingular matrix?

A nonsingular matrix is a square matrix that has a non-zero determinant, meaning it is invertible and has a unique solution.

2. How do you prove that a matrix is nonsingular?

To prove that a matrix A is nonsingular, you can show that the determinant of A is non-zero. This can be done through various methods such as finding the inverse of A or using the determinant formula.

3. What is the definition of the transpose of a matrix?

The transpose of a matrix A is a new matrix where the rows and columns are interchanged. This means that the first row of A becomes the first column of the transpose, the second row becomes the second column, and so on.

4. How do you prove that the transpose of a nonsingular matrix is also nonsingular?

To prove that the transpose of a nonsingular matrix A is also nonsingular, we can use the property that the determinant of the transpose is equal to the determinant of the original matrix, and since A is nonsingular, its determinant is non-zero, making the determinant of the transpose also non-zero.

5. What is the significance of proving that if A is nonsingular, then the transpose of A is also nonsingular?

Proving this statement is significant because it shows that the invertibility of a matrix is not affected by taking its transpose. This means that if we have a system of equations represented by a nonsingular matrix A, we can also solve it by using the transpose of A without changing the result.

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