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det (A^2+A+xI_2) = x

Source: Romanian District Olympiad 2006, Grade 11, Problem 1

March 11, 2006
linear algebramatrixalgebrapolynomiallinear algebra unsolved

Problem Statement

Let x>0x>0 be a real number and AA a square 2×22\times 2 matrix with real entries such that det⁡(A2+xI2)=0\det {(A^2+xI_2 )} = 0. Prove that det⁡(A2+A+xI2)=x\det{ (A^2+A+xI_2) } = x.
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