GATE 2025 ST – Question 26

Matrix Theory · Eigenvalues and Matrix Decompositions · 1 mark · Numerical answer

$P=\begin{pmatrix}1&2\\-1&4\end{pmatrix}$ and Q=P³−2P²−4P+13I₂. Find det Q (integer).

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Correct answer: 50 to 50

Explanation

P has trace 5, determinant 6 and eigenvalues 2,3. Under the polynomial q(t)=t³−2t²−4t+13 these become q(2)=5,q(3)=10 for Q. Their product is **det Q=50**.

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