GATE 2021 EE – Question 48
Let $A$ be a $10\times10$ matrix such that $A^5$ is a null matrix, and let $I$ be the $10\times10$ identity matrix. The determinant of $A+I$ is ________.
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Correct answer: 1
Explanation
If $A^5=0$ then every eigenvalue $\lambda$ of $A$ satisfies $\lambda^5=0$, so all eigenvalues are 0. The eigenvalues of $A+I$ are then all 1, so $\det(A+I)=1$.