## Determinant of matrices [closed]

Let $A$, $B$ and $C$ be matrices in $M_{11}(\mathbb{Z}_{97})$. Is there exists a matrix $D$ such that $det(A-D)=0$, $det(B-D)=0$ but $det(C-D)=-1$?

Is there exists a matrix $E$ such that $det(A-E)=0$, $det(B-E)=0$ but $det(C-E)=1$?

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Try thinking about a simple way, given any matrix $A$, to produce a matrix $D$ such that $\det(A-D)=0$. Then read the faq to find a more suitable website for this question. – Gerry Myerson Apr 10 2011 at 6:11