Understand the problem

True or false: Let A be a 3 \times 3 real symmetric matix s.t A^6=I. Then A^2=I
Source of the problem
TIFR 2018 Part A, Problem 6
Linear Algebra
Difficulty Level
Suggested Book
Linear Algebra, Hoffman and Kunze

Start with hints

Do you really need a hint? Try it first!

First investigate the properties of Real Symmetric Matrices.
The most important property of a real symmetric matrix A is encoded in “Spectral Decomposition” of A.
If matrix A then there exists Q with Q'Q=I such that A = Q'BQ,where B is a diagonal matrix with diagonal entries being the eigenvalues of A which are real numbers.
Here assume that the eigen values are a,b,c.
We will use this spectral decomposition of A.
Observe that A^6 = I \Rightarrow Q'(B^6)Q = I.(Check!)
Q'(B^6)Q=I \Rightarrow Q[Q'(B^6)Q]Q'= QQ'= I \Rightarrow B^6=I . [as Q is orthogonal]
B^6 is a diagonal matrix with diagonal entries real numbers raised to the power 6 i.e a^6,b^6 and c^6.
Now hint 3 implies a^6=1,b^6=1 and c^6=1.
a^2=1,b^2=1 and c^2=1 as a,b and c are real numbers. This means B^2=I.
A^2 = Q'(B^2)Q = Q'Q = I.
Hence the given statement is TRUE.

Watch the video

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

Radius of Convergence of a Power series | IIT JAM 2016

Try this problem from IIT JAM 2017 exam (Problem 48) and know how to determine radius of convergence of a power series.We provide sequential Hints.

Eigen Value of a matrix | IIT JAM 2017 | Problem 58

Try this problem from IIT JAM 2017 exam (Problem 58) and know how to evaluate Eigen value of a Matrix. We provide sequential hints.

Limit of a function | IIT JAM 2017 | Problem 8

Try this problem from IIT JAM 2017 exam (Problem 8). It deals with evaluating Limit of a function. We provide sequential hints.

Gradient, Divergence and Curl | IIT JAM 2014 | Problem 5

Try this problem from IIT JAM 2014 exam. It deals with calculating Gradient of a scalar point function, Divergence and curl of a vector point function point function.. We provide sequential hints.

Differential Equation| IIT JAM 2014 | Problem 4

Try this problem from IIT JAM 2014 exam. It requires knowledge of exact differential equation and partial derivative. We provide sequential hints.

Definite Integral as Limit of a sum | ISI QMS | QMA 2019

Try this problem from ISI QMS 2019 exam. It requires knowledge Real Analysis and integral calculus and is based on Definite Integral as Limit of a sum.

Minimal Polynomial of a Matrix | TIFR GS-2018 (Part B)

Try this beautiful problem from TIFR GS 2018 (Part B) based on Minimal Polynomial of a Matrix. This problem requires knowledge linear algebra.

Definite Integral & Expansion of a Determinant |ISI QMS 2019 |QMB Problem 7(a)

Try this beautiful problem from ISI QMS 2019 exam. This problem requires knowledge of determinant and definite integral. Sequential hints are given here.

What is TIFR and how to prepare for it?

About TIFR Tata Institute of Fundamental Research, TIFR is the foremost institution for advanced research in foundational sciences based in Mumbai, Maharashtra, India. The institute offers a master's course, an integrated M.Sc and Ph.D. course and a Ph.D. degree in...

Limit of a Sequence | IIT JAM 2018 | Problem 2

Try this beautiful problem from IIT JAM 2018 which requires knowledge of Real Analysis (Limit of a Sequence). We provide sequential hints.