Matrix solving problem

Question

I have a linear system like this: $$\left[\begin{array}{l}x_2\\y_2\end{array}\right] = \left[\begin{array}{l}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right]\left[\begin{array}{l}x_1\\y_1\end{array}\right]$$ $A_{11},A_{12},A_{21},A_{22}$ are known. The target is to get the matrix: $$\left[\begin{array}{l}y_1\\y_2\end{array}\right] = \left[\begin{array}{l}B_{11}&B_{12}\\B_{21}&B_{22}\end{array}\right]\left[\begin{array}{l}x_1\\x_2\end{array}\right]$$. For 2-by-2 matrix, I can solve it manually. But now I have a 4-by-4(6-by-6) matrix which I have to use Mathematica to get answer. Any suggestions on this problem?

Answer 1

Clear["Global`*"]

(Format[#[n__]] := Subscript[#, Row[{n}]]) & /@ {a, x, y};

eqns = Thread[{x[2], y[2]} == Array[a, {2, 2}] . {x[1], y[1]}]

sol = Collect[Solve[eqns, {y[1], y[2]}][[1]], {x[1], x[2]}]

eqns /. sol // Simplify

(* {True, True} *)

B = List @@@ Values[sol] /. {x[1] -> 1, x[2] -> 1}