GATE 2021 ST – Question 23

Matrix Theory · Eigenvalues and Matrix Decompositions · 1 mark · Numerical answer

A real 2×2 matrix has eigenvalues 1,−1 with corresponding eigenvectors (√3/2,1/2)ᵀ and (−1/2,√3/2)ᵀ. If A²⁰²¹ has entries a,b,c,d,find a+b+c+d,to two decimals.

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Correct answer: 1.70 to 1.75

Explanation

The two eigenvectors are orthonormal,and odd powers preserve the eigenvalues 1,−1,so A²⁰²¹=A. Its spectral expansion vvᵀ−wwᵀ gives A=[[1/2,√3/2],[√3/2,−1/2]]. Entry sum=√3≈**1.73**.

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