Consider the matrix A=bmatrix&x\\&bmatrix. If its eigenvalues are +j and -j, the unknown x equals

GATE 2016 EC (EC3) – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Consider the matrix
$$A=\begin{bmatrix}\sigma&x\\\omega&\sigma\end{bmatrix}.$$
If its eigenvalues are $\sigma+j\omega$ and $\sigma-j\omega$, the unknown x equals

$+j\omega$
$-j\omega$
$+\omega$
$-\omega$

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Correct answer: (D) $-\omega$

Explanation

The characteristic polynomial is $(\lambda-\sigma)^2-x\omega$. The given roots instead require $(\lambda-\sigma)^2+\omega^2$.

Equating constants gives $x\omega=-\omega^2$, so **x=−ω, D**, for the intended nonzero ω. If ω=0, the eigenvalues alone do not fix x; D remains the keyed expression for the nondegenerate case.

GATE 2016 EC (EC3) – Question 11

Explanation

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