\[(B`B)=\left(\begin{array}{c}-12 &-4 &1 &-2 \\ 0 &4 &3 &-2\end{array}\right)\]

Using Gauss-Jordan elimination we obtain the above in the form \(\displaystyle{\left({I}{P}^{{-{1}}}\right)}\)

\[(IP^{-1})=\left(\begin{array}{c}1 &0 &-1/3 &1/3 \\0 &1 &3/4 &-1/2\end{array}\right)\]

Then, the transition matrix form B to B’ is:

\[P^{-1}=\left(\begin{array}{c}-1/3 &1/3\\ 3/4 &-1/2\end{array}\right)\]