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