A=pmatrix0&2\\2&0pmatrix. On M₂(ℂ), T(B)=AB. Find T’s characteristic polynomial.

GATE 2024 MA – Question 19

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

$A=\begin{pmatrix}0&2\\2&0\end{pmatrix}$. On M₂(ℂ), T(B)=AB. Find T’s characteristic polynomial.

$X^4-8X^2+16$
$X^2-4$
$X^2-2$
$X^4-16$

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Show answer and explanation

Correct answer: (A) $X^4-8X^2+16$

Explanation

Left multiplication acts independently as A on each of B’s two columns. A has eigenvalues 2,−2, so T has each twice. Polynomial=(X−2)²(X+2)²=X⁴−8X²+16. **A**.

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