SOLUTION: Suppose A is an invertible n × n matrix. Must the system of equations A x = x have a unique solution? Explain your reasoning.

Question 1164526: Suppose A is an invertible n × n matrix. Must the system of equations A x = x have a unique solution? Explain your reasoning.
Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.

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 in the coordinate plane with
the rotation angle different from 0 (from zero or from any multiple of the full angle ).

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