GATE 2021 EC – Question 46
A real $2\times2$ non-singular matrix $A$ with repeated eigenvalue is given as
$$A=\begin{bmatrix}x&-3.0\\3.0&4.0\end{bmatrix}$$
where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 10
Explanation
A repeated eigenvalue $\lambda$ means the characteristic polynomial is a perfect square, so the discriminant is zero: $(\text{trace})^2-4\det=(x+4)^2-4(4x+9)=x^2-8x-20=0$. Hence $x=10$ or $x=-2$, and the positive root is $x=10$.