$A×x×B=C$ What is x?

Question

$$\left(\begin{matrix}-1&0\\1&1\\\end{matrix}\right)\cdot X\cdot \left(\begin{matrix}1&2\\0&1\\\end{matrix}\right)=\left(\begin{matrix}0&1\\1&-1\\\end{matrix}\right)$$ How can I solve this matrix equation?

Answer 1

Hint:

You have a matrix equation: $$ AXB=C $$ with $ A$ and $ B$ invertible matrices, so you can find: $$ A^{-1}AXBB^{-1}=X=A^{-1}CB^{-1}$$

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$$ A^{-1}CB^{-1}= \begin{bmatrix} -1&0\\1&1 \end{bmatrix} \begin{bmatrix} 0&1\\1&-1 \end{bmatrix} \begin{bmatrix} 1&-2\\0&1 \end{bmatrix} = \begin{bmatrix} 0&-1\\1&0 \end{bmatrix} \begin{bmatrix} 1&-2\\0&1 \end{bmatrix} = \begin{bmatrix} 0&-1\\1&-2 \end{bmatrix} $$