Question 1164526
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No.



The matrix equation   Ax = x   means that the matrix   A   has an eigenvalue equal to  1.



Far not every square invertible matrix  A  has eigenvalue  1.



A contradictory example is any  2x2-matrix of the rotation by angle  {{{alpha}}}   in the coordinate plane with

the rotation angle   {{{alpha}}}   different from   0 (from zero or from any multiple of the full angle  {{{2pi}}}).



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Solved, answered and explained.