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