MHB Inverse of adjoint - where is my mistake ?

  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Inverse Mistake
Yankel
Messages
390
Reaction score
0
Hello all, I have a matrix A:

\[\begin{pmatrix} 2 &4 &1 \\ -4 &7 &3 \\ 5 &1 &-2 \end{pmatrix}\]

and I need to find the adjoint of the matrix inverse.

I found adj(A) to be:

\[\begin{pmatrix} -17 &9 &5 \\ 7 &-9 &-10 \\ -39 &18 &30 \end{pmatrix}\]

and I found the determinant of A to be -45 and the determinant of adj(A) to be 2025.

Now based on:

\[adj(A^{-1})=(adj(A))^{-1}\]

I tried solving the question, I did:

\[B=adj(A))\]

and looked for:

\[B^{-1}\]

This way:

\[B^{-1}=\frac{1}{\left | B \right |}adj(B)\]

and got:

\[\frac{1}{2025}A\]

which is not the answer. the answer should be:

\[-\frac{1}{45}A\]

And I don't understand what I did wrong here.

Thank you
 
Physics news on Phys.org
Yankel said:
Hello all, I have a matrix A:

\[\begin{pmatrix} 2 &4 &1 \\ -4 &7 &3 \\ 5 &1 &-2 \end{pmatrix}\]

and I need to find the adjoint of the matrix inverse.

I found adj(A) to be:

\[\begin{pmatrix} -17 &9 &5 \\ 7 &-9 &-10 \\ -39 &18 &30 \end{pmatrix}\]

and I found the determinant of A to be -45 and the determinant of adj(A) to be 2025.

Now based on:

\[adj(A^{-1})=(adj(A))^{-1}\]

I tried solving the question, I did:

\[B=adj(A))\]

and looked for:

\[B^{-1}\]

This way:

\[B^{-1}=\frac{1}{\left | B \right |}adj(B)\]

and got:

\[\frac{1}{2025}A\]

which is not the answer. the answer should be:

\[-\frac{1}{45}A\]

And I don't understand what I did wrong here.

Thank you

Hi Yankel,

The inverse of $A$ is given by:
$$A^{-1}=\frac{1}{\det A} \text{adj }A$$
See adjugate matrix (as it is called with less ambiguity) on wiki.Speaking about $\det(\text{adj }A)$, it relates to $\det A$ as:
$$\det(\text{adj }A) = (\det A)^{n-1} = (-45)^{3-1} = 2025$$
 
I like Serena, thank you !

among the formulas out there I mentioned that I did find det(adj(A)) to be 2025, it was easy to miss this line.

This is not what I am asking. Taking the first formula you mentioned, I found the adj(inverse of A), and I was wrong, and can't find my mistake.
 
Yankel said:
I like Serena, thank you !

among the formulas out there I mentioned that I did find det(adj(A)) to be 2025, it was easy to miss this line.

This is not what I am asking. Taking the first formula you mentioned, I found the adj(inverse of A), and I was wrong, and can't find my mistake.

We have:
$$A^{-1} = \frac 1{\det A} \text{adj }A$$
Therefore we also have:
$$A = \frac 1{\det A^{-1}} \text{adj}(A^{-1}) \quad\Rightarrow\quad \text{adj}(A^{-1}) = \det A^{-1} \cdot A = \frac 1{\det A} \cdot A$$

Yankel said:
Now based on:

\[adj(A^{-1})=(adj(A))^{-1}\]

I tried solving the question, I did:

\[B=adj(A))\]

and looked for:

\[B^{-1}\]

This way:

\[B^{-1}=\frac{1}{\left | B \right |}adj(B)\]

Let's substitute $B=\text{adj}(A)$. That gives us:

\[B^{-1}=\frac{1}{\left | \text{adj}(A) \right |}\text{adj}(\text{adj}(A))\]

Did you evaluate $\text{adj}(\text{adj}(A))$?
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
Back
Top