Consider two hypothetical matrices A and B. These matrices are defined as follow:
$$ A = \left( \begin{array}{ccc} 0 & a & b & c\\ 0 & 0 & 0 & d \\ 0 & 0 & 0 & e \\ 0 & 0 & 0 & 0 \end{array} \right) $$
$$ B = \left( \begin{array}{ccc} 0 & 1-(1-b)(1-c) & 1-(1-a)(1-c) & 1-(1-a)(1-b)\\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array} \right) $$
My question: Is it possible to find an algebraic formula in which matrix B can be calculated (or achieved) from matrix A?
Thank you.