TIFR GS 2019, Part A Problem 2

Linear Algebra

Easy

Linear Algebra: Hoffman and Kunze

Do you really need a hint? Try it first!

\(A^2+A+1=A(A+I)+I=0\). 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.

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.