Question 69
Let $A$ be a singular real matrix of order $n\ge2$, and let $A^*$ denote its adjugate. Show that $\operatorname{tr}(A^*)\ne-1$ if and only if $I_n+A^*$ is invertible.
Second source statement (Hungarian version): The singular square matrix $A$ of order $n$ has real entries, $n\ge2$. Let $A^*$ be its adjugate. Prove that $I_n+A^*$ is invertible if and only if $\operatorname{tr}(A^*)\ne-1$.
Second source statement (Hungarian version): The singular square matrix $A$ of order $n$ has real entries, $n\ge2$. Let $A^*$ be its adjugate. Prove that $I_n+A^*$ is invertible if and only if $\operatorname{tr}(A^*)\ne-1$.