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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?

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3 Answers

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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.

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yes up until the last step. the inverse needs to be multiplied to the right of the matrix B

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\begin{align} & & EA & = B \\[10pt] \text{right: } & & (EA)A^{-1} & = BA^{-1} \\ & & E(A A^{-1}) & = BA^{-1} \\ & & E & = BA^{-1} \\[12pt] \text{wrong: } & & A^{-1}(EA) & = A^{-1}B \end{align}

In most cases, $A^{-1} E A$ is not equal to $E$.

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