There are two $3\times3$ matrices $A$ and $B.$ Both represent a rotation in 3D space. $A$ and $B$ are given as follows where $a,b,c$ are column vectors.
$A = [\begin{array}{ccc}a & b & c \\ \end{array}]$
$B = [\begin{array}{ccc}a & c & -b \\ \end{array}]$
The $3$rd column of each matrix is the cross product of the $1$st and $2$nd column. How are these matrices related if in any way?
On
If you think about the matrices as linear transformations, you can see what their inverses do without having to compute their entries. Namely, $A^{-1}a=\hat x$, $A^{-1}b=\hat y$, and $A^{-1}c = \hat z$. Similarly, $B^{-1}a=\hat x$, $B^{-1}b=-\hat z$, and $B^{-1}c = \hat y$.
Now you can determine the products $A^{-1}B$ and $B^{-1}A$ by trying out what they do to $\hat x$, $\hat y$, and $\hat z$. Both products should turn out to be $90^\circ$ rotations around $\hat x$, in opposite directions. Similarly, both $AB^{-1}$ and $BA^{-1}$ are $90^\circ$ rotations around $a$.
Another way we could write that relationship is $$ B = A \, \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{bmatrix} $$ I'm not sure if that fully answers your question though...