I got the following matrix equation
$AX^{-1} - AB = AC^{T}$
Where A is invertable. In step two the key moves from the above equation to:
$X^{-1} - B = C^{T}$
But as I recalled we cannot just "remove" a matrix from the equation. Is this step legal only when the matrix removed is in all terms on both sides?
Lets say I instead had
$AX^{-1} - AB = AC^{T} + X$
Would this stop me from removing A from both sides, since A is not a part of every term anymore?
Cheers!
On
This works because $A$ is invertible so it has an inverse say $A^{-1}$.
Then we can just multiply both sides on the left by $A^{-1}$.
$A^{-1}(AX^{-1}-AB)=A^{-1}AC^T$, as $A^{-1}$ is the inverse of $A$ we get $X^{-1} - B = C^T$
We can always do this, let's do the same to $AX^{-1}-AB=AC^T + X$
$A^{-1}(AX^{-1}-AB)=A^{-1}(AC^T + X)$
$X^{-1}-B=C^T + A^{-1}X$
In the first instance when you are "removing a matrix", what you are really doing is multiplying on the left by $A^{-1}$, which exists since $A$ is invertible.
In the second instance, if we multiply on the left by $A^{-1}$, we get $$ X^{-1} - B = C^{T} + A^{-1}X $$