Inverse of adjoint - where is my mistake ?

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Discussion Overview

The discussion revolves around the calculation of the adjoint of the inverse of a matrix, specifically addressing the matrix A provided by the participants. The focus is on the mathematical properties and relationships involving the adjoint and determinant of matrices, as well as the implications of these properties in finding the adjoint of the inverse.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a matrix A and calculates its adjoint and determinant, leading to an expression for the adjoint of the inverse.
  • Another participant confirms the relationship between the adjoint of a matrix and its determinant, stating that the determinant of the adjoint is related to the determinant of the original matrix.
  • There is a challenge regarding the calculation of the adjoint of the inverse, with one participant expressing confusion over their results and seeking clarification on where their mistake lies.
  • Participants discuss the formula for the adjoint of the inverse and the implications of the determinant in the calculations.
  • There is a suggestion to evaluate the adjoint of the adjoint, which may be relevant to resolving the confusion expressed by one participant.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach to finding the adjoint of the inverse, and there are competing views on the calculations and interpretations of the properties involved.

Contextual Notes

There are unresolved aspects regarding the evaluation of the adjoint of the adjoint and the implications of the determinant in the context of the calculations presented. The discussion reflects uncertainty about the correct application of the formulas and relationships among the matrices involved.

Yankel
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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
 
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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 $\text{adj}(\text{adj}(A))$?
 

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