Balance a matrix equation

Question

I want to know how to do operations on a balanced equations of matrices, because matrix multiplication is no commutative. My problem is $$AX{A}^{-1}=B$$ on the left side of the equation I want to multiply by $A^{-1}$, how would that look on the right side. My operation: $${(A^{-1}})AX{A}^{-1}=B$$, on the left side would that look like $$A^{-1}B$$ or $$BA^{-1}$$

Answer 1

When handling matrices you must do operations on both sides, on the same side of the expression.

For example, in considering the similarity relation $ A = PDP^{-1}$, if I want to right-multiply by $P$, I'll get

$$ AP = PDP^{-1}P $$

which reduces to

$$ AP = PD $$

(exactly what it means for $A$ to be similar to $D$). Notice how I right-multiplied (also known as post-multiplying) by the same object on the right side of the equals sign.

Answer 2

If you left-multiply the LHS you must left-multiply also the RHS, so we have: $$ AXA^{-1}=B \iff A^{-1}AXA^{-1}=A^{-1}B \iff XA^{-1}=A^{-1}B $$