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GATE 2021 EE – Question 11

Engineering Mathematics · Linear Algebra: Eigen values, Eigen vectors · 1 mark · Multiple choice

Let $p$ and $q$ be real numbers such that $p^2+q^2=1$. The eigenvalues of the matrix $\begin{bmatrix}p&q\\q&-p\end{bmatrix}$ are

  1. 1 and 1
  2. 1 and $-1$
  3. $j$ and $-j$
  4. $pq$ and $-pq$

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Correct answer: (B) 1 and $-1$

Explanation

The trace is $p-p=0$ and the determinant is $-p^2-q^2=-1$. So the characteristic equation is $\lambda^2-1=0$, giving eigenvalues $1$ and $-1$.