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

[sihag]

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 ?

 

[daveyinaz]

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.

 

[transgalactic]

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