I came across this in a problem:
$$\frac{\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}+\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}}{1+\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}}$$
Then, in order to simplify this, I thought the following:
\begin{gather} \frac{\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}+\mathbf{u}\color{red}{\left(\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}\right)}\mathbf{v}^{T}\mathbf{A}^{-1}}{1+\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}}\\ \frac{\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}+\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}\color{red}{\left(\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}\right)}}{1+\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}}\\ \frac{\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}\left(1+\color{red}{\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}}\right)}{1+\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}}\\ \mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1} \end{gather}
I figured that this could be done because of associativity and as the terms in $\color{red}{\text{red}}$ result in a scalar, it could be 'moved around'. Is this correct? If yes, does this always hold or are there any conditions on the vectors and matrices involved?