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I am trying to see if there is a "nice" ( relatively straightforward) way of finding the

solution/kernel of the map : ##f(A)=A^n -Id ## , where A is an ## k \times k ## matrix and ##n## is a positive

integer. Question: what is the kernel of this map? Cranking out matrix coefficients gets messy

and very slow as n grows. I can think of a solution using the Fundamental Theorem of Linear

Algebra relating Ortho spaces of rows, columns and row- and column- spaces. Is there some other

"reasonable" way of finding all solutions (clearly Id,-Id are solutions for n even and Id is a solution for n odd

)?