I am asked to find a matrix $E$ such that $EA= B$.
I am given matrix $A$ which is $4\times 4$ and matrix $B$ $4\times4$.
Would I find $E$ the following way or is incorrect?
$$EA=B$$
$A^{-1} [EA = B]$ Multiply by $A^{-1}$ on both sides
$E = BA^{-1}$.
E = $A^{-1} B$ (Not sure if this step is correct by matrix multiplication)
So, therefore I would find matrix $E$ by finding the inverse of $A$ and then multiplying it by matrix $B$? Is that correct?
Best Answer
In general, if matrices $$A=C$$ then $$AB=CB \tag{right multiplication}$$ and $$BA = BC \tag{left multiplication}$$ The order is important since matrix multiplication isn't commutative. So, from $$EA=B $$$$\implies (EA)(A^{-1})=B(A^{-1}) \tag{right multiplication}$$$$ \implies E(AA^{-1}) = BA^{-1} \tag{associativity of multiplication} $$ $$\implies E = BA^{-1} \tag{as $AA^{-1}=I$ and $EI = E$}$$
Here, however we have assumed that an inverse exists ($A^{-1}$), which might not be true in general.