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