GATE 2025 ST – Question 57

Matrix Theory · Eigenvalues and Matrix Decompositions · 2 marks · Numerical answer

$P=\begin{pmatrix}2&1\\1&2\end{pmatrix}$, $Q=\begin{pmatrix}1&1\\-2&4\end{pmatrix}$. Find trace(P⁵+Q⁴) (integer).

Show answer and explanation

Correct answer: 341 to 341

Explanation

P has eigenvalues 3,1, so trace P⁵=3⁵+1=244. Q has trace 5,determinant 6, hence eigenvalues 2,3; trace Q⁴=2⁴+3⁴=97. Sum=**341**.

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