The matrix $AB = C$ where $A$, $B$ and $C$ are all $2 \times 2$ non-singular matrices.
How would I go about to solve for the Matrix $A$ and express it in terms of $B$ and $C$?
There are two methods I found but they give different answers.
Method 1:
Multiply both sides on the left by $B^{-1}$ so $B^{-1}(AB)=B^{-1}C$
Then rewrite as $B^{-1}BA=B^{-1}C$ since multiplication is associative
$IA=B^{-1}C$ therefore $A=B^{-1}C$
However the answer is different if instead I multiply on the right by $B^{-1}$ which is what I would have done instead (and is the correct answer).
Method 2:
$(AB)B^{-1}=CB^{-1}$
$AI=CB^{-1}$
$A=CB^{-1} $
My question is, why is it that method 1 is incorrect? What is the mistake and could someone elaborate on the general rules I should follow to solve questions like this.
Multiplication of matrices is associative, but NOT commutative, so $B^{-1}AB$ is not the same as $B^{-1}BA$.
Here's a counterexample to commutativity:
$$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 3 & 0 \end{pmatrix} \neq \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$