 TIFR 2013 Problem 8 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Linear Algebra by Gilbert Strang. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta

## Question:True/False?

If a real square matrix $$A$$ is similar to a diagonal matrix and satifies $$A^n=0$$ for some $$n$$, then $$A$$ must be the zero matrix.

Hint: There exists an invertible matrix $$P$$ and a diagonal matrix $$D$$ which satisfies $$PDP^{-1}=A$$. What happens when we apply the given condition?

## Discussion:

$$0=A^n=(PDP^{-1})^n=PDP^{-1}PDP^{-1}…PDP^{-1}$$ (n-times multiplication)

Hence, $$0=PD^nP^{-1}$$. $$P$$ being invertible, we multiply on left and right by $$P^{-1}$$ and $$P$$ respectively and get $$D=0$$.

In whatever basis you write the zero transformation, the result is same, namely the matrix of zero transformation is always zero-or null matrix.

Hence, $$A=0$$

you can also see the-TIFR 2014 Problem 11 Solution – Nilpotent Matrix Eigenvalues

## Helpdesk

• What is this topic: Linear Algebra
• What are some of the associated concept: Invertible Matrix, Linear Transformation
• Book Suggestions: Introduction to Linear Algebra by Gilbert Strang