
Matrix equation
Given the matrix equation
$\displaystyle X^{1}A^{T} + B^{1}C = 0$
Where $\displaystyle A,B,C,$ and $\displaystyle X$ are 2 x 2 invertible matrices. Solve for $\displaystyle X$ in terms of $\displaystyle A,B,C$.
ii) Find X when $\displaystyle A = \begin{matrix} [1 & 1] \\ [0 & 2] \end{matrix}, B= \begin{matrix} [1 & 2] \\ [1 & 1] \end{matrix}, C= \begin{matrix} [1 & 2] \\ [1 & 4] \end{matrix}$

Re: Matrix equation
$\displaystyle x=\left(\begin{array}{cc} 1 & 3 \\ 1 & 2\end{array}\right)$(Shake)

Re: Matrix equation
For solving for X (part i) of the question , is this correct?
$\displaystyle B^{1} C =  X^{1} A^{T}$
left multiply by X
$\displaystyle X B^{1} C =  (X X^{1}) A^{T}$
$\displaystyle X B^{1} C =  I A^{T}$
right multiply by $\displaystyle C^{1} B$
$\displaystyle X B^{1} C (C^{1} B) =  A^{T} (C^{1} B)$
$\displaystyle X =  A^{T} C^{1} B$
as
$\displaystyle B^{1} C (C^{1} B) = B^{1} (C C^{1}) B =$
$\displaystyle = B^{1} I B = B^{1} B = I$

Re: Matrix equation
Yes, of course that is correct.

Re: Matrix equation
Thanks HallsofIvy. The problem now is that when I plug in my 2x2 matrices to the equation $\displaystyle X = A^{T} C^{1} B$ , I do not get the matrix that MaxJasper provided above.

Re: Matrix equation
Typo: Sorry, minus sign was dropped, so correct x is:
$\displaystyle \text{x}=\left(\begin{array}{cc} 1 & 3 \\ 1 & 2\end{array}\right)$(Nod)

Re: Matrix equation