## IMC 2014 Day 1 Problem 1

Determine all pairs of real numbers for which there exists a unique symmetric matrix with real entries satisfying and .

**IMC 2014 Day 1 Problem 1**

**Solution:** First note that the matrix must be diagonal, since

Have the same trace and determinant.

If we have a diagonal matrix then has the same trace and determinant. For it to be unique we would need that and therefore . This means that and and the characteristic polynomial of is .

Conversely, if then has both eigenvalues equal to . Since is symmetric, is diagonalizable and there exists a matrix such that . This last relation implies that and the matrix is unique.

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Categories: Olympiad, Problem Solving
Algebra, IMC

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