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I was wondering if a matrix $A$ with dimensions $m \times n$ has a property (similar to transposition) where we define a new matrix $B$ with: $$ b_{ij} = a_{i(n-j+1)} $$ For example if we have: $$ A = \begin{pmatrix}3 & 2 & 1 \\ 6 & 1 & 4 \\ 4 & 0 & 1\end{pmatrix} $$ then: $$ B =\begin{pmatrix}1 & 2 & 3 \\ 4 & 1 & 6 \\ 1 & 0 & 4\end{pmatrix} $$

Does the matrix $B$ (given some $A$ that isn't the zero matrix) have any properties that are the same as in $A$? One can for example tell that $\text{Col } A = \text{Col } B$, but what about determinants? Is $B$ invertible if $A$ is?

And if so, why?

L0Ludde0
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  • We will have $|\det(A)|=|\det(B)|$ assuming $m=n$ , also $rank(A)=rank(B)$. I think, there are more properties. – Peter Oct 02 '20 at 08:04
  • If $m=n$, we also have that $A$ is invertible if and only if $B$ is. – Peter Oct 02 '20 at 08:06

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