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GATE 2022 MA – Question 65

Linear Algebra · Eigenvalues and Canonical Forms · 2 marks · Numerical answer

A real 3×3 M satisfies M²=2M+3I and det M=−9. Find trace M.

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Correct answer: 5 to 5

Explanation

Every eigenvalue satisfies λ²−2λ−3=(λ−3)(λ+1)=0. If k eigenvalues equal 3, determinant is 3^k(−1)^(3−k). Value −9 requires k=2, so eigenvalues are 3,3,−1 and trace is **5**.

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