Find the matrix A that satisfies the equation Find the matrix A that satisfies the equation
$$
\left[
2\left[
\begin{array}{}
  1&1\\
  -2&3
\end{array}
\right] -5A^{-1}
\right]
^T= 
\left(4A^T
\right)^{-1}$$
I'm not sure how to start solving this question. where/how do i start and how do i think? 

Let the given matrix $$\begin{pmatrix} 1 & 1 \\ -2 & 3 \end{pmatrix}=B$$ and call $(A^T)^{-1}=C=(A^{-1})^T$, then we have $$(2B^T-5(A^{-1})^T)=(4A^T)^{-1} \implies 2B^T-5C= \frac{1}{4}C\implies B^T=\frac{21}{8} C \implies (B^T)^T=\frac{21}{8} C^T=\frac{21}{8} A^{-1} \implies B=\frac{21}{8} A^{-1} \implies A= \frac{21}{8}B^{-1}$$
$$\implies A=\frac{21}{40}\begin{pmatrix} 3 &-1 \\ 2 & 1 \end{pmatrix}$$
Here we have used the interchangeability of transpose and inverse.
