Are There Relationships Between the Column Vectors of a Matrix and Its Inverse?

[srfriggen]

Are there any special relationships between the column vectors of a matrix and the corresponding column vectors of its inverse?

 

[chiro]

Are there any special relationships between the column vectors of a matrix and the corresponding column vectors of its inverse?

Hey sfriggen.

Are you talking about the most general non-singular matrix or a specific class of non-singular matrices?

For example rotation matrices have the property that the tranpose of the matrix is the inverse which means that the inverse matrix of a rotation matrix has the columns as the rows of the original.

 

[srfriggen]

I'm asking for any invertible matrix.

 

[AlephZero]

Why should there be any relationship?

Look at the simplest example. The inverse of $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ is $1 / (ad - bc)\begin{bmatrix}\phantom{-}d & -b \\ -c & \phantom{-}a\end{bmatrix}$

There's no relatioship between the columns that I can see. There is a relationship between the ROWS of the inverse and the columns of the matrix, of course.

 

[srfriggen]

Thanks AlephZero,

I wasn't implying there should be any relationship, just asking if there was one.

How could one generalize the 2x2 example you showed?

 

[AlephZero]

You could write the inverse in terms of cofactors. That certainly shows each column depends on every element of the matrix in a rather complicated way.

But if you want to "prove" there is no relationship, the first problem is trying to define what you mean by "any possible sort of relationship". You can't prove something doesn't exist unless you can define it somehow.