Question

How would I go about proving that if a matrix is invertible, then so is its adjoint?

Answer 1

There are a couple of ways. One would involve taking the ajoint of the matrix equation:\[A^{-1}A=I\iff (A^{-1}A)^H=I^H\iff A^H(A^{-1})^H=I\] so because there is a matrix i can multiply A^H by to get the identity, its invertible.

Answer 2

Most times when we talk about invertible matrices, we are talking about square matrices. So if A is invertible, that means it has full rank, and its columns space and row space have the same rank. The row space of A is the column space of A^H, so the matrix A^H also has full rank, which means its invertible.

Answer 3

that's right, I am only having to show this for the special case of a 2 x 2 matrix, also, I am not familiar with the superscript H

Answer 4

replace H with T's if you are talking about matrices with only real entries.

Answer 5

H is for Hermitian, which has the same function as transpose, except if your matrices have complex entries, its the complex conjugate transpose.

Answer 6

basically, write down what i put in my first post, but change the H to T's, and your fine.

Answer 7

ok... I am still digesting that answer now. The hint that was given to us involves the formula \[A^{-1}=adj(A)/\left| A \right|\] and I have worked that out to \[1/ad-bc \left[\begin{matrix}d & -c\\ -b & a\end{matrix}\right]\]

Answer 8

oh oh oh, thats my bad. i was thinking transposes instead. Ignore what i said then. You want to go about by using that equation and multiplying both sides by the matrix A. You get:\[A^{-1}=\frac{1}{|A|}adj (A)\iff AA^{-1}=\frac{1}{|A|}A\cdot adj(A)\]\[\iff I=\frac{1}{|A|}A\cdot adj(A)\]So because there is a matrix i can multiply adj(A) by to get the identity, it is invertible.

Answer 9

The idea is that if you can work with a matrix and somehow get something in the form:\[AB=I\] then both A and B are invertible, and they are each other's inverses.

Answer 10

ok... I need to look at that a few minutes, but some pieces are starting to fall into place now.

Answer 11

Awesome... I see that now, and if I want to prove the converse, then I really just need to do the same thing in reverse.

Answer 12

yes sir :)

Answer 13

very cool. I really appreciate your help. I like the pi shirt btw :) I'm in the midst of writing a 15 page paper on the history of pi lol.

Answer 14

oh wow, then you know way more about the number than I do!

Answer 15

I dunno about that, but found out a lot of cool things about it. Hopefully 15 pages worth ;) Anyway, I'm going to get back on this homework. Going to keep this site up though... I think I found a new internet hangout lol. Again, thanks so much for your help :)