Why matrices are invertible

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. Simple answer : because by definition a matrix is commutative with its inverse on multiplication.

More complicated answer : There exists a left inverse and a right inverse that is defined for all matrices including non-square matrices. Since this question has just been bumped anyway and I feel like I have something else to add, here are my thoughts:. The answer is no:. By the dimension formula we have:. Maybe I am bumping this but I want to add an example for clarity.

This can only be possible with square matrices. But, this proof could be augmented with geometric intuition I guess. Here is an example that shows how non-square matrices harm the invertibility of a linear transformation.

But as all linear transformations, it maps the zero vector to zero vector. As you can see, this is just a counterexample but I think, someone should prove that non-square matrices can not have inverses because they can not uniquely map. I think proof has got to do with linear dependence and independence. After this step, we will manipulate this transformation to get to linearly independent vectors. I hope you will be able to see how this generalizes. Add a comment. Active Oldest Votes.

DonAntonio DonAntonio k 17 17 gold badges silver badges bronze badges. Sigur Sigur 5, 1 1 gold badge 23 23 silver badges 43 43 bronze badges. Another quick question: You can find the inverse using both the determinant AND gauss jordan elimation, right? What you are describing is a 1-sided inverse.

Sign up or log in Sign up using Google. Sign up using Facebook. Learn more. Ask Question. Asked 3 days ago. Active 3 days ago. Viewed 99 times. I'm surely not understanding the connection between functions and matrices, can someone clarify?

After this definition and an example It is stated the theorem I mentioned, is it wrong? Tortar Tortar 3, 2 2 gold badges 8 8 silver badges 22 22 bronze badges. The statement you quote looks very suspicious - can you provide more context for the quote. Are square matrices always invertible? Notations: Note that, all the square matrices are not invertible.

If the square matrix has invertible matrix or non-singular if and only if its determinant value is non-zero. Do all matrices have inverses? Note: Not all square matrices have inverses. A square matrix which has an inverse is called invertible or nonsingular, and a square matrix without an inverse is called noninvertible or singular. Does the zero matrix have an inverse? If the determinant of the matrix is zero, then it will not have an inverse; the matrix is then said to be singular.

Only non-singular matrices have inverses.