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**nirax** consider $\displaystyle A, A^2, A^3, ..., A^{n^2}, A^{{n^2}+1}, ... $ if this set spans the full space then you must get $\displaystyle {n^2} $ lin independent ones among these. the highest power occuring in such a set must be at least $\displaystyle {n^2} $ if you try to write an expression of the form $\displaystyle a_1.A + a_2.A^2 + a_3. A^3 + ... + a_i.A^{i} $, this sum would never be zero unless you take $\displaystyle i$ to be atleast $\displaystyle {n^2} $ by the earlier argument of lin independence.

so the minimal polynomial of this matrix has degree at least $\displaystyle {n^2} $, contradicting the fact that minimal poly cannot be of degree higher than $\displaystyle {n} $.

which step you do not follow ??