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I would like to find a "mechanic" way in order to solve such questions.

Find a matrix $A \in \mathbb{R}^{3\times3}$ corresponding to the following:

$ A\cdot A=$ \begin{pmatrix} 1 & 0 &2 \\ 0 &1 &0 \\ 0& 0 &1 \end{pmatrix}

Obviously its \begin{pmatrix} 1 & 0 &1 \\ 0 &1 &0 \\ 0& 0 &1 \end{pmatrix}.

But it was more or less a guess. Is there a way or rule or something involving $A^{-1}$ or gaussian elimination or something else which allows to solve for more complex matrices given such question?

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    $\begingroup$ Do you know the jordan canonical form? $\endgroup$ – Daniel Apr 6 '15 at 18:21
  • $\begingroup$ if you can find the eigenvalues and eigenvectors, i.e., spectral decomposition of $A^2,$ then you can find an $A$ to fit the given $A^2.$ $\endgroup$ – abel Apr 6 '15 at 18:25
  • $\begingroup$ If the eigenvalues of a real matrix B are real then the jordan canonical form is very useful to find A such that p(A)=B, where p(x) is a polynomial. See the following answer for an example: math.stackexchange.com/questions/990392/… $\endgroup$ – Daniel Apr 6 '15 at 18:26
  • $\begingroup$ Unfortunately, Im not familiar with these concepts. However, thank you for your suggestions!! $\endgroup$ – Mamba Apr 6 '15 at 19:15

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