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GATE 2025 MA – Question 23

Linear Algebra · Eigenvalues and Canonical Forms · 1 mark · Multiple select

For which invertible M is the difference of characteristic polynomials p_M(x)−p_(M⁻¹)(x) constant?

  1. $\begin{pmatrix}5&7\\2&3\end{pmatrix}$
  2. $\begin{pmatrix}3&1\\4&2\end{pmatrix}$
  3. $\begin{pmatrix}1&3\\2&-1\end{pmatrix}$
  4. $\begin{pmatrix}5&-8\\2&-3\end{pmatrix}$

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Correct answer: (A) $\begin{pmatrix}5&7\\2&3\end{pmatrix}$; (C) $\begin{pmatrix}1&3\\2&-1\end{pmatrix}$; (D) $\begin{pmatrix}5&-8\\2&-3\end{pmatrix}$

Explanation

For 2×2 matrices, trace(M⁻¹)=trace(M)/det(M). The x coefficient cancels iff tr(M)=tr(M)/det(M), meaning trace zero or determinant one. A has determinant 1, B determinant 2 and trace 5, C trace 0, D determinant 1. **A,C,D**.

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