Answer
$$
P^{-1}=\left[ \begin {array}{cccccc} 0&-4&4\\
-1&-6&8\\ 3&14&-12\end {array} \right].
$$
Work Step by Step
Given
$$
B=\{(1,2,4),(-1,2,0),(2,4,0)\}, \quad B^{\prime}=\{(0,2,1),(-2,1,0),(1,1,1)\}.
$$
To find the transition matrix from $B$ to $B^{\prime}$, we form the matrix
$$
\left[B^{\prime} B\right]= \left[ \begin {array}{cccccc} 0&2&1&1&2&4\\ -2&1&0&
-1&2&0\\ 1&1&1&2&4&0\end {array} \right]
.$$
Using Gauss-Jordan elimination to obtain the transition matrix
$$
\left[\begin{array}{ll}{I_{3}} & {P^{-1}}\end{array}\right]= \left[ \begin {array}{cccccc} 1&0&0&0&-4&4\\ 0&1&0&
-1&-6&8\\ 0&0&1&3&14&-12\end {array} \right]
.$$
So, we have
$$
P^{-1}=\left[ \begin {array}{cccccc} 0&-4&4\\
-1&-6&8\\ 3&14&-12\end {array} \right].
$$
