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Real Symmetric matrix: TIFR 2018 Part A, Problem 6

This problem is a cute and simple application of the real symmeric matrices in the linear algebra section. It appeared in TIFR GS 2018.

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
Topic
Linear Algebra
Difficulty Level
Medium
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.

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