## Elementary Linear Algebra 7th Edition

$$P^{-1}=\left[ \begin {array}{cccc} 6&-1\\ -5&1 \end {array} \right] .$$
Given $$B=\{(1,0),(0,1)\}, B^{\prime}=\{(1,1),(5,6)\} .$$ To find the transition matrix from $B$ to $B^{\prime}$, we form the matrix $$\left[B^{\prime} B\right]=\left[ \begin {array}{cccc} 1&1&1&0\\ 5&6&0&1 \end {array} \right] .$$ Using Gauss-Jordan elimination, the reduced row echelon form is $$\left[\begin{array}{ll}{I_{2}} & {P^{-1}}\end{array}\right]=\left[ \begin {array}{cccc} 1&0&6&-1\\ 0&1&-5&1 \end {array} \right] .$$ So, we have $$P^{-1}=\left[ \begin {array}{cccc} 6&-1\\ -5&1 \end {array} \right] .$$