Prove Invertibility of Square Matrix A & ATA

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In summary, a square matrix A is invertible if and only if its transpose multiplied by itself (ATA) is also invertible. This can be proven by showing that if A is invertible, then det(AAT) is non zero, and if det(AAT) is non zero, then A is invertible as well. This can be done by using the property that the determinant of a product of matrices is equal to the product of their determinants, and the fact that the determinant of an invertible matrix is non zero.
  • #1
iamsmooth
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Homework Statement


Prove that asquare matrix A is invertible if nad only if ATA is invertible


Homework Equations


Hmm, can't think of any. A A-1 = I maybe?


The Attempt at a Solution



I have trouble with these theory questions, so I'm not sure how to approach this.

If something is invertible, that means it's determinant is not 0? So does it have something to do with that?

Thanks :eek:
 
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  • #2


If you aren't good at "theory" questions, then try rephrasing them as "calculation" questions!

If something is invertible, that means it's determinant is not 0?
And vice versa as well, right? If the determinant is not zero, then it's invertible. So we have an identity:
{X is invertible} = {det X is not zero}​

What happens if you substitute this into the original question? (There are two things to substitute! Do both!) Is it something that you can solve by doing two calculations?
 
  • #3


A square matrix A is invertible if and only if ATA is invertible

Well, If A is invertible, then det(A) is not 0.

det(A) = det(AT)

The poduct of two invertible matrices is invertible

So therefore ATA is invertible if and only if A is invertible.

This doesn't seem like proof, nor does it seem coherent :( I suck
 
  • #4


You have the right idea but try to show both directions separately.
If A is invertible, then det(A) is non zero. So what can you say about det(AAT)?
Similarly, if det(AAT) is non zero, what can you say about det(A)?
(Also remember that det(AB) = det(A)det(B))
 

1. What is a square matrix?

A square matrix is a matrix with an equal number of rows and columns. It is represented by the notation An x n, where n is the number of rows and columns.

2. What does it mean for a matrix to be invertible?

A matrix is invertible if it has an inverse matrix, denoted as A-1, that when multiplied together, results in the identity matrix (I). This means that the inverse of a matrix A, A-1, can "undo" the operations performed by A, resulting in the identity matrix.

3. How do you prove the invertibility of a square matrix A?

To prove the invertibility of a square matrix A, you must show that A has an inverse matrix A-1. This can be done by using various methods such as the Gauss-Jordan elimination, the determinant method, or the adjoint method.

4. What is the significance of proving invertibility of a square matrix A?

Proving the invertibility of a square matrix A is important because it guarantees that the matrix has a unique solution for its system of linear equations. It also allows for the efficient computation of matrix operations, such as finding the determinant, inverse, and eigenvalues.

5. What is ATA and how does it relate to proving the invertibility of a square matrix A?

ATA stands for A transpose multiplied by A. It is used to determine the invertibility of a square matrix A by checking its eigenvalues. If all the eigenvalues of ATA are positive, then A is invertible. This is because the eigenvalues of ATA are the squares of the singular values of A, and a matrix is invertible if and only if its singular values are all non-zero.

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