a question about transpose and inverse of matrix

Question

I have a question about transpose and inverse of matrix.

Let $\boldsymbol X$ and $\boldsymbol Y$ be any matrices.

Please calculate this expression step by step: $$\left[\left(\boldsymbol{X}^\mathsf{T}\boldsymbol{Y}\right)^{-1}\right]^\mathsf{T}$$ where ${}^\mathsf{T}$ is transpose.

Answer:

$$\left[\left(\boldsymbol X^\mathsf{T}\boldsymbol Y\right)^{-1}\right]^\mathsf{T}=\left[\left(\boldsymbol X^\mathsf{T}\boldsymbol Y\right)^\mathsf{T}\right]^{-1}=\left[\boldsymbol Y^\mathsf{T}\boldsymbol X\right]^{-1}=\boldsymbol X^\mathsf{T}\boldsymbol Y$$

Is this right?

Answer 1

You'll use that $(AB)^{-1} = B^{-1}A^{-1}$, $(AB)^T = B^TA^T$ and $(A^{-1})^T = (A^T)^{-1}$:

\begin{equation} \begin{split} ((X^T Y)^{-1})^T & = (Y^{-1}(X^T)^{-1})^T \\ & = (Y^{-1}(X^{-1})^T)^T \\ & = ((X^{-1})^T)^T(Y^{-1})^T \\ & = X^{-1} (Y^T)^{-1} \end{split} \end{equation}

Answer 2

Why $(Y^TX)^{-1} = X^TY$? It should be $(Y^TX)^{-1} = X^{-1}(Y^T)^{-1}$.