GATE 2024 EE – Question 32
The sum of the eigenvalues of the matrix $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}^2$ is ______ (rounded off to the nearest integer).
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Correct answer: 29
Explanation
The sum of the eigenvalues is the trace. $\begin{bmatrix}1&2\\3&4\end{bmatrix}^2=\begin{bmatrix}7&10\\15&22\end{bmatrix}$, whose trace is $7+22=29$. (Check: eigenvalues of the matrix satisfy $\lambda_1+\lambda_2=5$, $\lambda_1\lambda_2=-2$, so $\lambda_1^2+\lambda_2^2=25+4=29$.)