Matrix equation

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October 19th 2012, 07:33 AM
dumbledore

Matrix equation

Given the matrix equation

$X^{-1}A^{T} + B^{-1}C = 0$

Where $A,B,C,$ and $X$ are 2 x 2 invertible matrices. Solve for $X$ in terms of $A,B,C$ .

ii) Find X when $A = \begin{matrix} [1 & 1] \\ [0 & 2] \end{matrix}, B= \begin{matrix} [1 & 2] \\ [1 & 1] \end{matrix}, C= \begin{matrix} [1 & 2] \\ [1 & 4] \end{matrix}$
October 19th 2012, 08:54 AM
MaxJasper

Re: Matrix equation

$x=\left(\begin{array}{cc} 1 & 3 \\ 1 & 2\end{array}\right)$ (Shake)
October 20th 2012, 06:33 AM
dumbledore

Re: Matrix equation

For solving for X (part i) of the question , is this correct?

$B^{-1} C = - X^{-1} A^{T}$

left multiply by X

$X B^{-1} C = - (X X^{-1}) A^{T}$

$X B^{-1} C = - I A^{T}$

right multiply by $C^{-1} B$

$X B^{-1} C (C^{-1} B) = - A^{T} (C^{-1} B)$

$X = - A^{T} C^{-1} B$

as

$B^{-1} C (C^{-1} B) = B^{-1} (C C^{-1}) B =$

$= B^{-1} I B = B^{-1} B = I$
October 20th 2012, 08:55 AM
HallsofIvy

Re: Matrix equation

Yes, of course that is correct.
October 20th 2012, 09:43 AM
dumbledore

Re: Matrix equation

Thanks HallsofIvy. The problem now is that when I plug in my 2x2 matrices to the equation $X = -A^{T} C^{-1} B$ , I do not get the matrix that MaxJasper provided above.
October 20th 2012, 12:55 PM
MaxJasper

Re: Matrix equation

Typo: Sorry, minus sign was dropped, so correct x is:

$\text{x}=-\left(\begin{array}{cc} 1 & 3 \\ 1 & 2\end{array}\right)$ (Nod)
October 20th 2012, 01:03 PM
dumbledore

Re: Matrix equation

Got it!

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