GATE 2023 MA – Question 35

Linear Algebra · Eigenvalues and Canonical Forms · 1 mark · Numerical answer

Let $M=\begin{pmatrix}3&-1&-2\\0&2&4\\0&0&1\end{pmatrix}$. If 6M⁻¹=M²−6M+αI, find α.

Show answer and explanation

Correct answer: 11 to 11

Explanation

The characteristic polynomial is (λ−3)(λ−2)(λ−1)=λ³−6λ²+11λ−6. Cayley–Hamilton and multiplication by M⁻¹ give M²−6M+11I−6M⁻¹=0. Therefore **α=11**.

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