Answer
$$
P^{-1}=\left[ \begin {array}{cccc} \frac{3}{5}&\frac{8}{5}\\ \frac{4}{5}&\frac{9}{5}
\end {array} \right].
$$
Work Step by Step
Given
$$
B=\{(2,5),(1,2)\}, B^{\prime}=\{(2,1),(-1,2)\}.
$$
To find the transition matrix from $B$ to $B^{\prime}$, we form the matrix
$$
\left[B^{\prime} B\right]=\left[ \begin {array}{cccc} 2&1&2&5\\ -1&2&1&2
\end {array} \right]
.$$
Using Gauss-Jordan elimination to obtain the transition matrix
$$
\left[\begin{array}{ll}{I_{2}} & {P^{-1}}\end{array}\right]=\left[ \begin {array}{cccc} 1&0&\frac{3}{5}&\frac{8}{5}\\ 0&1&\frac{4}{5}&\frac{9}{5}
\end {array} \right]
.$$
So, we have
$$
P^{-1}=\left[ \begin {array}{cccc} \frac{3}{5}&\frac{8}{5}\\ \frac{4}{5}&\frac{9}{5}
\end {array} \right].
$$
