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True or False: Let $A \in M_{10}(\Bbb R)$ satisfies $A^2+A+I$, then $A$ is invertible.
Source of the problem
TIFR GS 2019, Part A Problem 2
Linear Algebra
Easy
Suggested Book
Linear Algebra: Hoffman and Kunze

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$$A^2+A+1=A(A+I)+I=0$$. Can you take it from here?
$$A^2+A+1=A(A+I)+I=0\Rightarrow A(-A-I)=I$$. So the inverse of $$A$$ is $$-A-I$$. Hence the statement is true.

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