The original matrix R from an inverse

Question

I this is exercise I'am given the inverse of the matrix R. I'am trying to find the original matrix R from the inverse R. How can I do that?

Thx, for any reply!

Answer 1

HINT

Recall that

$$(A^{-1})^{-1}=A$$

thus we need to evaluate the inverse of $R^{-1}$ for example by Gauss-Jordan

$$[R^{-1}\quad I]\to [I\quad R]$$

Answer 2

We suppose that $R$ is $n \times n$. Let us denote the columns of $R$ by $c_1,...c_n$ and let $e_j=(0,...,0,1,0,,,,0)$ ($1$ in the j-the place).

Then solve the linear systems

$R^{-1}x=e_j$

($j=1,...,n)$.

For the unique solution x of the equation $R^{-1}x=e_j$ we have $x=s_j$.