GATE 2022 MA – Question 11

Linear Algebra · Eigenvalues and Canonical Forms · 1 mark · Multiple choice

M∈ℂ³ˣ³ has characteristic polynomial λ³+αλ²+βλ−1, with α+β≠0. Which identity holds?

M(I−βM)=M⁻¹(M+αI)
M(I+βM)=M⁻¹(M−αI)
M⁻¹(M⁻¹+βI)=M−αI
M⁻¹(M⁻¹−βI)=M+αI

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Show answer and explanation

Correct answer: (D) M⁻¹(M⁻¹−βI)=M+αI

Explanation

The constant term ensures M is invertible. Cayley–Hamilton gives M³+αM²+βM−I=0. Multiply by M⁻²: M+αI+βM⁻¹−M⁻²=0. Rearrangement yields **M⁻¹(M⁻¹−βI)=M+αI, D**.

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