GATE 2021 IN – Question 48
For $A=\begin{bmatrix}2&5\\0&3\end{bmatrix}$, the determinant $\det(A^4-5A^3+6A^2+2I)$ is ______.
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Show answer and explanation
Correct answer: 4 to 4
Explanation
A is upper triangular with diagonal eigenvalues 2 and 3. A polynomial in A remains upper triangular with diagonal p(2),p(3). Here p(2)=16−40+24+2=2 and p(3)=81−135+54+2=2. Their product is the determinant, **4**.