I have these matrices given:
$$A= \begin{bmatrix} 3&1&1&1\\1&2&0&{-1}\\0&1&2&{-1}\\0&0&1&1\end{bmatrix}$$
$$T= \begin{bmatrix} 1&0&0\\0&1&0\\0&0&{-1}\\0&0&1\end{bmatrix}$$
$$C= \begin{bmatrix} 1&2&3\\0&1&2\\0&0&1\end{bmatrix}$$
And I have to find matrix $B$ for which the following equation is true: $${ABC=4AT+2BC}$$
so far I managed to get this: $${B=4AT(A-2I)^{-1}C^{-1} }$$
But I got a problem with $AT$, because it cannot be multiplicated. Help would be really appreciated.
Do the matrix mathematics: $$ABC=4AT+2BC$$ Pass $2BC$ to the left-hand side, preserving order (because it matters): $$ABC-2BC=4AT$$ Factor out the $BC$ because matrix multiplication does not commute: $AB \neq BA$ $$(A-2I)BC=4AT$$ Right-multiply $C^{-1}$ to remove $C$ from the left-hand side: $$(A-2I)B=4ATC^{-1}$$ Left-multiply $(A-2I)^{-1}$ to remove $(A-2I)$ from the left-hand side: $$B=(A-2I)^{-1}4ATC^{-1}$$