Inverse of adjoint - where is my mistake ?

In summary, the conversation discusses finding the adjoint of a matrix inverse, with a given matrix A and its adjoint matrix. The determinant of A and its adjoint are also mentioned. The formula for finding the adjoint of a matrix inverse is given, and the conversation explores the use of this formula to solve the problem. The mistake made by the speaker is pointed out and a corrected formula is provided.
  • #1
Yankel
395
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
 
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  • #2
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:
$$\det(\text{adj }A) = (\det A)^{n-1} = (-45)^{3-1} = 2025$$
 
  • #3
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.
 
  • #4
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 $\text{adj}(\text{adj}(A))$?
 
  • #5


It seems like you have made a mistake in calculating the adjoint of A. The correct adjoint matrix should be:

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

Notice that the signs in the second row are different from what you have calculated. This mistake leads to the incorrect answer you obtained for B^{-1}. Double check your calculations for the adjoint matrix and try again.
 

FAQ: Inverse of adjoint - where is my mistake ?

What is the inverse of adjoint?

The inverse of adjoint refers to the inverse of the matrix that is obtained by taking the transpose of the cofactor matrix of a given square matrix. It is also known as the adjugate matrix.

How do I find the inverse of adjoint?

To find the inverse of adjoint, you first need to calculate the determinant of the original matrix. Then, you can find the adjugate matrix by taking the transpose of the cofactor matrix. Finally, divide the adjugate matrix by the determinant to obtain the inverse of the adjoint.

Why is the inverse of adjoint important?

The inverse of adjoint is important because it allows us to find the inverse of a non-invertible matrix. This is useful in solving linear systems of equations and in finding the inverse of large matrices.

Can there be any mistakes while finding the inverse of adjoint?

Yes, there can be mistakes while finding the inverse of adjoint. Some common mistakes include miscalculating the determinant or the cofactor matrix, or making errors while taking the transpose. It is important to double check the calculations to ensure accuracy.

How can I check if I have made a mistake while finding the inverse of adjoint?

The best way to check for mistakes is to multiply the original matrix with the calculated inverse of adjoint. If the result is the identity matrix, then the inverse of adjoint has been calculated correctly. If not, then there may be a mistake in the calculations.

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