0% Complete
0/59 Steps

Invertible Matrix implies identity?: TIFR GS 2019, Part B Problem 4

Understand the problem

Is it possible to have a non-identity 2 \times 2 diagonalizable, invertible, complex matrix A s.t characteristics polynomials of A and A^2 are the same?

Source of the problem
TIFR GS 2019, Part B Problem 4
Topic
Linear algebra
Difficulty Level
Hard
Suggested Book
Linear Algebra; Hoffman and Kunze

Start with hints

Do you really need a hint? Try it first!

Think in terms of eigenvalues. Let a and b are eigenvalues of A then eigenvalues of A^2 are a^2 and b^2.
You need t^2-(a+b)t+ab=t^2-(a^2+b^2)t+a^2b^2 So ab=a^2b^2\implies ab=1 as A^{-1} exists. Can you think from here? What about a+b and a+1/a?

a+b=a^2+b^2\implies a+1/a=a^2+1/a^2\implies a^4-a^3-a+1=0 \implies (a-1)(a^3-1)=0\implies a=1,\omega,\omega^2  

Now, what are the cases?

Discard a=1 as A\ne I and diagonalizable. Discard the case of both eigenvalues equal as A is not diagonalizable for repeated eigenvalues. The only possibility is \{\omega,\omega^2\} as the eigenvalue set for A. So A=\begin{bmatrix}\omega&0\\0&\omega^2\end{bmatrix}.   Hence, the statement is false.

Watch the video (Coming Soon)

Connected Program at Cheenta

College Mathematics Program

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.

Similar Problems

What is the 30th term in the Fibonacci series?

We all know about this sequence , but is this really possible to find the 30th term with just adding the previous trems in order to get such large terms?

What’s the speciality of the number ‘5040’?

I have posted the answer on quora of this question. The number 5040 has many special number theoretic aspects and I have listed all of this one by one.

4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 2

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

4 questions from Sylow’s theorem: Qn 1

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

Arithmetical Dynamics: Part 0

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points . Theorem: Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at...

Sum based on Probability – ISI MMA 2018 Question 24

This is an interesting and cute sum based on the concept of Arithematic and Geometric series .The problem is to find a solution of a probability sum.

System of the linear equation: ISI MMA 2018 Question 11

This is a cute and interesting problem based on System of the linear equation in linear algebra. Here we are finding the determinant value .

Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .