Question

Solve the given matrix equation for X. Simplify your answers as much as possible. (In the words of Albert Einstein, "Everything should be made as simple as possible, but not simpler.") Assume that all matrices are invertible. AXB = (BA)2

Answer 1

The solution is
`X=A^(-1)(BA)^(2)B^(-1)`.

Explanation:

Given two matrix
`A, B` are invertible, so they are not commutative. That is,
`AB\\eq BA`. Now given,

`AXB=(BA)^(2)`

`A^(-1)AXB=A^(-1)(BA)^(2)`

`XB=A^(-1)(BA)^(2)`

`XBB^(-1)=A^(-1)(BA)^(2)B^(-1)`

`X=A^(-1)(BA* BA)^(2)B^(-1)`

Since invertible matrices are non-commutative, in the next step we cannot write
`BA` as
`AB`. And so the required answer is,

`X=A^(-1)(BA)^(2)B^(-1)`.