# Understand the problem

True or false: Let be a real symmetric matix s.t . Then

##### 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 then there exists with such that ,where is a diagonal matrix with diagonal entries being the eigenvalues of A which are real numbers.

Here assume that the eigen values are .

We will use this spectral decomposition of A.

Observe that .(Check!)

. [as Q is orthogonal]

is a diagonal matrix with diagonal entries real numbers raised to the power 6 i.e and .

Now hint 3 implies and .

and as and are real numbers. This means .

.

Hence the given statement is TRUE.

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