Is my solution of this matrix equation correct ?
$(AX^{-1}+B)^{-1}=\frac13X$, I've started off by raising the equation to $-1$ and in the end I get : $((\frac13I-A)^{-1}B)^{-1}$
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Not exactly, it should be 3I instead 1/3*I. Also, you can simplify your solution a bit. – user2097 Aug 23 '15 at 17:34
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\begin{eqnarray} (AX^{-1}+B)^{-1} & = & \frac{1}{3}X\\ AX^{-1}+B & = & 3X^{-1}\\ B & = & (3I-A)X^{-1}\\ BX & = & (3I-A)\\ X & = & B^{-1}(3I-A) \end{eqnarray}
Dr Xorile
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You need to be a little careful to left multiply and right multiply correctly. It's a right multiply by $X$ in the second last line, and left multiply by $B$ in the last line. – Dr Xorile Aug 23 '15 at 17:49