Overview of Matrix Invertibility: How Can We Prove That a Matrix is Invertible?

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In summary, the conversation discusses the proof that matrix A is invertible if and only if its columns and rows are linearly independent. This can be proven by using linear transformations or by performing a row reduction on the matrix and checking for a non-zero determinant. The importance of using the specific method of proof required by the professor is also mentioned.
  • #1
sihag
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Suppose A : n x n
A is invertible iff the columns (and rows) of A are linearly independent

A is invertible
iff det |A| is non-zero
iff rank A is n
iff column rank is n
iff dim (column space is n)
iff the n columns of A are linearly independent

Well, this is a proof that I laid down. It was junked by my prof. She said I have to use linear transformations to prove it. Someone throw some light ?
 
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  • #2
I would suggest looking up umm...other...methods of proof. Such as when you see an "iff", you might want to try and prove that if A is invertible, then the columns and rows of A are linearly independent. Then try to prove that if the rows and columns within A are linearly independent, then A is invertible.

Also, sometimes questions want you to prove it a certain way...so while you might be able to prove it another way, using other methods or something..it's not necessarily relevant to the topic of discussion...I really don't know since I'm not in your class. Hope this helps.
 
  • #3
in order to prove matrix invertibility
thereare two ways
the ony is to make a row reduction on this matrix
and if in the end of the process you don't have a line of zeros
then its invertable.

the other way is if the determinant of this metrix differs zero then its invertable
 

Related to Overview of Matrix Invertibility: How Can We Prove That a Matrix is Invertible?

What is a matrix invertibility question?

A matrix invertibility question refers to the process of determining whether a given matrix can be inverted, or converted into its inverse matrix. This involves checking if the matrix is square and if its determinant is non-zero.

Why is the invertibility of a matrix important?

The invertibility of a matrix is important because it allows us to solve systems of linear equations, which have many real-world applications. It also helps us to find the inverse of a matrix, which can be useful in various mathematical operations.

How do you determine if a matrix is invertible?

To determine if a matrix is invertible, we need to check if it is a square matrix (same number of rows and columns) and if its determinant is non-zero. If both of these conditions are met, then the matrix is invertible.

What happens if a matrix is not invertible?

If a matrix is not invertible, it is also known as a singular matrix. This means that it does not have an inverse matrix and cannot be used to solve systems of linear equations. It is also not possible to perform certain mathematical operations on a singular matrix.

Can a non-square matrix be invertible?

No, a non-square matrix cannot be invertible because it does not have an equal number of rows and columns, and therefore does not have a determinant. However, a rectangular matrix can be invertible if it has linearly independent columns.

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