how do I find matrix 3x3 which is non-diagonalizable and fits this function
$$A^{-1}=A^2+A-I$$
where I is the identity matrix.
I started by multiplying it by $A$ so
$$I=A^3+A^2-A$$
and finally
$$0=A^3+A^2-A-I$$
Now I know that fits it's own characteristic polynomial, which is what my problem is, how do I find a matrix with characteristic polynomial
$$λ^3+λ^2-λ-1=0 ??$$ And also how do I incorporate the fact that $A$ is non-diagonalizable.
Thank you.
The roots of this polynomial are $-1$ (twice) and $1$ (once). So, take$$A=\begin{pmatrix}1&0&0\\0&-1&1\\0&0&-1\end{pmatrix}.$$