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Is there a general formula, or a specific technique to find the inverse matrix of matrices where the entries are variables instead of numbers (is it even possible or defined)? For example, how does one find the inverse of: $ \left( \begin{array}{ccc} e^x\cos y & -e^x \sin y \\ e^x \sin y & e^x\cos y \\ \end{array} \right)$ or $ \left( \begin{array}{ccc} 2xy & 4 \\ y & x^2 \\ \end{array} \right)$

[If these examples don't work for some reason, are there matrices with variables where an inverse can be found?]

Thanks in advance!

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    $\begingroup$ So the formula $A^{-1} = \frac{1}{|A|}$adj A still works? What about the row elimination method? $\endgroup$ – Gabriel Feb 10 '15 at 1:44
  • $\begingroup$ @Gabriel Multiplying $A$ by the matrix of cofactors is already what elimination does, without doing divisions. The result is a diagonal matrix with $det(A)$ in the diagonal. $\endgroup$ – Carol Feb 10 '15 at 1:51
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The inverse of a Vandermonde matrix can be found. It is related to the Lagrange interpolation polynomial.

Also try the inverse of $$ ( \binom{j}{i} t^{j-i})_{1\le i, j \le n}$$ a Pascal triangle with weights, representing the shift operator on a space of polynomials.

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