If [tex]\mathbf A^4=\mathbf I[/tex], then the characteristic polynomial of [tex]\mathbf A^4[/tex] is
[tex]p_{\mathbf A^4}(\lambda)=\det(\lambda\mathbf I-\mathbf A^4)=\det((\lambda-1)\mathbf I)=(\lambda-1)^4[/tex]
which means [tex]\mathbf A^4[/tex] has eigenvalues [tex]\lambda=1[/tex].
We know that if [tex]\chi[/tex] is an eigenvalue of [tex]\mathbf X[/tex], then [tex]\chi^n[/tex] is an eigenvalues of [tex]\mathbf X^n[/tex].
So if [tex]\lambda=1[/tex] is the only eigenvalue of [tex]\mathbf A^4[/tex], we know that [tex]\pm\sqrt[4]\lambda=\pm1[/tex] are the only possible eigenvalues of [tex]\mathbf A[/tex]. We can construct five possible characteristic polynomials for [tex]\mathbf A[/tex] in that case.
[tex]p_{\mathbf A}(\lambda)=(\lambda-1)^4[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)(\lambda-1)^3[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)^2(\lambda-1)^2[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)^3(\lambda-1)[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)^4[/tex]
